One checks that $L$ is a closed, densely defined linear operator and that for every $f \in \mathcal{D}(L)$ the map $t \mapsto T_t f$ is continuously differentiable with

$$ \frac{d}{dt} T_t f = T_t L f = L T_t f, \quad t \ge 0. \qquad (9.5) $$

Thus $(T_t)$ is the unique (mild) solution operator of the abstract Cauchy problem

$$ \left\{ \begin{array}{l} \frac{du}{dt}(t) = Lu(t), \quad t > 0, \\ u(0) = f, \end{array} \right. \qquad u(t) = T_t f. \qquad (9.6) $$

Two fundamental generation results characterize which closed operators arise as generators of $C_0$-semigroups. We state them in the forms most used in the theory of linear evolution equations on Hilbert spaces.

**Theorem 9.1 (Hille-Yosida (Hilbert-space form)).** A linear operator $(L, \mathcal{D}(L))$ on $\mathcal{H}$ is the generator of a strongly continuous semigroup $(T_t)_{t \ge 0}$ with $\|T_t\| \le Me^{\omega t}$ if and only if $L$ is closed, densely defined, and for every $\alpha > \omega$ the resolvent $(\alpha I - L)^{-1}$ exists as a bounded operator on $\mathcal{H}$ and satisfies the uniform resolvent estimate

$$ \|(\alpha I - L)^{-n}\| \le \frac{M}{(\alpha - \omega)^n}, \quad n \in \mathbb{N}. \qquad (9.7) $$

*Proof.* We prove the Hilbert-space version of the Hille-Yosida theorem in two directions. All limits and integrals below are taken in the strong (i.e. norm) topology of the Hilbert space $\mathcal{H}$ unless otherwise stated.

Let $(T_t)_{t \ge 0}$ be a strongly continuous semigroup on $\mathcal{H}$ with the growth bound $|T_t| \le Me^{\omega t}$ for all $t \ge 0$. Denote by $L$ its infinitesimal generator. First we show necessity: $L$ is closed, densely defined, and the resolvent estimates hold.

Because $(T_t)$ is a $C_0$-semigroup, its generator $L$ is by standard semigroup theory densely defined and closed; moreover $\mathcal{D}(L)$ is dense in $\mathcal{H}$. For $\alpha > \omega$ define the operator

$$ R(\alpha) := \int_{0}^{\infty} e^{-\alpha t} T_t dt, \qquad (9.8) $$

where the Bochner integral converges in operator norm because

$$ |e^{-\alpha t} T_t| \le Me^{-(\alpha-\omega)t} \qquad (9.9) $$

and

$$ \int_{0}^{\infty} e^{-(\alpha - \omega)t}, dt = (\alpha - \omega)^{-1} < \infty \qquad (9.10) $$

Differentiating under the integral sign (justified by the uniform exponential bound) yields

$$ (\alpha I - L)R(\alpha) = \int_{0}^{\infty} e^{-\alpha t} (\alpha I - L)T_t dt = \int_{0}^{\infty} e^{-\alpha t} \frac{d}{dt}(T_t), dt = [-e^{-\alpha t} T_t]_{t=0}^{t=\infty} + \alpha \int_{0}^{\infty} e^{-\alpha t} T_t dt = I. \qquad (9.11) $$

A similar calculation shows $R(\alpha)(\alpha I - L) = I$. Hence $R(\alpha) = (\alpha I - L)^{-1}$ exists and is bounded
for every $\alpha > \omega$. Moreover

$$ |(\alpha I - L)^{-1}| = |R(\alpha)| \le \int_0^\infty e^{-\alpha t} |T_t|, dt \le M \int_0^\infty e^{-(\alpha-\omega)t}, dt = \frac{M}{\alpha-\omega}. \quad (9.12) $$

To obtain the stated power-norm estimates, use the Laplace representation for higher resolvent powers. For every integer $n \ge 1$ and $\alpha > \omega$ we may write (Bochner integral, iterated)

$$ (\alpha I - L)^{-n} = \frac{1}{(n-1)!} \int_0^\infty t^{n-1} e^{-\alpha t} T_t, dt, \quad (9.13) $$

which is obtained by successive differentiation of the scalar Laplace transform or by convolution powers of the kernel. Taking norms and using the growth bound one gets

$$ |(\alpha I - L)^{-n}| \le \frac{1}{(n-1)!} \int_0^\infty t^{n-1} e^{-\alpha t} |T_t|, dt \le \frac{M}{(n-1)!} \int_0^\infty t^{n-1} e^{-(\alpha-\omega)t}, dt = \frac{M}{(n-1)!} \frac{(n-1)!}{(\alpha-\omega)^n} = \frac{M}{(\alpha-\omega)^n} \quad (9.14) $$

This proves the necessity.

We now prove sufficiency: assume $L$ is a closed densely defined operator and that for some constants $M \ge 1$, $\omega \in \mathbb{R}$ and every $\alpha > \omega$ the resolvent $(\alpha I - L)^{-1}$ exists on $\mathcal{H}$ and satisfies

$$ |(\alpha I - L)^{-n}| \le \frac{M}{(\alpha - \omega)^n}, \quad n \in \mathbb{N}. \quad (9.15) $$

We will construct a $C_0$-semigroup $(T_t)_{t \ge 0}$ with $|T_t| \le Me^{\omega t}$ and generator $L$.

For each fixed $\alpha > \omega$ define the bounded (Yosida) approximant

$$ L_{\alpha} := \alpha^{2}(\alpha I - L)^{-1} - \alpha I = \alpha L(\alpha I - L)^{-1}. \quad (9.16) $$

One checks algebraically, using

$$ (\alpha I - L)^{-1} L = \alpha (\alpha I - L)^{-1} - I \quad (9.17) $$

that $L_\alpha$ is a bounded operator on $\mathcal{H}$. The resolvent bounds give the uniform estimate (for $\alpha > \omega$)

$$ |L_{\alpha}| \le \alpha^2 |(\alpha I - L)^{-1}| + \alpha \le \frac{\alpha^2 M}{\alpha - \omega} + \alpha < \infty, \quad (9.18) $$

so $L_\alpha$ generates the uniformly continuous semigroup $e^{tL_\alpha}$ on $\mathcal{H}$. We shall show that the family $T_t^\alpha := e^{tL_\alpha}$ converges strongly, as $\alpha \downarrow \infty$, to a $C_0$-semigroup $T_t$ whose generator is $L$.

Two standard and equivalent constructions may be used; we present the one via Laplace transforms (resolvents) because it yields clean operator-norm estimates and the growth bound directly.

Fix $\lambda > \omega$. For each $\alpha > \lambda$ compute the resolvent of $L_\alpha$. Using the algebraic identity be-
tween resolvents one obtains (straightforward algebra; details follow)

$$ (\lambda I - L_{\alpha})^{-1} = (\lambda I - L)^{-1}(I + (L_{\alpha} - L)(\lambda I - L_{\alpha})^{-1}), \qquad (9.19) $$

and, more usefully for our purpose, one derives the explicit relation

$$ (\lambda I - L_{\alpha})^{-1} = (\lambda I - L)^{-1} (I - (\alpha - \lambda)(\lambda I - L)^{-1})^{-1}. \qquad (9.20) $$

From the hypothesis

$$ |(\alpha I - L)^{-1}| \leq M/(\alpha - \omega) \qquad (9.21) $$

and the choice $\lambda > \omega$ we may choose $\alpha$ large enough that

$$ |(\alpha - \lambda)(\alpha I - L)^{-1}| < 1 \qquad (9.22) $$

hence the Neumann series is valid and $(\lambda I - L_{\alpha})^{-1}$ is well defined and bounded uniformly in $\alpha$ for $\alpha$ large. A direct computation (using the resolvent bounds and the Neumann series summation) yields the estimate

$$ |(\lambda I - L_{\alpha})^{-1}| \leq \frac{M}{\lambda - \omega} \cdot \frac{1}{1 - (\alpha - \lambda) \frac{M}{\alpha - \omega}} \leq C(\lambda, M, \omega) \qquad (9.23) $$

for $\alpha$ sufficiently large; in particular the resolvents $(\lambda I - L_{\alpha})^{-1}$ are uniformly bounded and converge strongly to $(\lambda I - L)^{-1}$ as $\alpha \to \infty$. Indeed, this strong convergence follows from the identity

$$ (\lambda I - L_{\alpha})^{-1} - (\lambda I - L)^{-1} = (\lambda I - L_{\alpha})^{-1}(L_{\alpha} - L)(\lambda I - L)^{-1}, \qquad (9.24) $$

combined with $\lim_{\alpha \to \infty} |(L_{\alpha} - L)u| = 0$ for each $u \in \mathcal{D}(L)$ (a standard property of Yosida approximants), and density of $\mathcal{D}(L)$.

Having established that the bounded-resolvent family $(\lambda I - L_{\alpha})^{-1}$ converges strongly to $(\lambda I - L)^{-1}$ for some (hence all) $\lambda > \omega$, we apply the Post-Widder inversion (or the Laplace inversion formula for resolvents) to pass to the semigroup level. For each $\alpha$ the semigroup $T_t^{\alpha} = e^{tL_{\alpha}}$ satisfies the Laplace representation

$$ (\lambda I - L_{\alpha})^{-1} = \int_{0}^{\infty} e^{-\lambda t} T_{t}^{\alpha} dt, \quad \lambda > |L_{\alpha}|. \qquad (9.25) $$

By dominated convergence in Bochner integrals (using uniform bounds on $|T_t^{\alpha}|$ to be established below) and the strong resolvent convergence we deduce that the family $(T_t^{\alpha})_{t \geq 0}$ converges strongly, for each fixed $t \geq 0$, to an operator $T_t$ characterized by

$$ (\lambda I - L)^{-1} = \int_{0}^{\infty} e^{-\lambda t} T_{t}, dt, \quad \lambda > \omega. \qquad (9.26) $$

The operator-valued Laplace transform inversion then ensures that $(T_t)_{t \geq 0}$ is a strongly continuous semigroup whose generator has resolvent $(\lambda I - L)^{-1}$ and therefore must coincide with $L$. We must verify the uniform bound $|T_t| \leq M e^{\omega t}$ and the strong continuity.

To obtain the growth bound, we use the uniform resolvent estimates. For each $\alpha > \omega$, the
bounded generator $L_\alpha$ produces a uniformly continuous semigroup $T_t^\alpha$ with

$$T_t^\alpha = \frac{1}{2\pi i} \int_\gamma e^{t\zeta} (\zeta I - L_\alpha)^{-1}, d\zeta, \qquad (9.27)$$

where $\gamma$ is a vertical contour in the complex plane to the right of the spectrum of $L_\alpha$. Using the resolvent estimates inherited from the hypothesis one obtains, after contour deformation and standard resolvent-semigroup estimates, that for all $t \ge 0$ and all $\alpha > \omega$,

$$|T_t^\alpha| \le Me^{\omega t}. \qquad (9.28)$$

One concrete way to see this is to use the inequality

$$| (\beta I - L_\alpha)^{-1} | \le M / (\beta - \omega) \qquad (9.29)$$

for $\beta > \omega$, which follows from the resolvent identities and the assumed bounds; then integrate the scalar bound

$$\int_0^\infty e^{-\beta t} |T_t^\alpha|, dt \le M / (\beta - \omega) \qquad (9.30)$$

and apply the Post-Widder inversion to obtain the multiplicative factor $Me^{\omega t}$.

Since $|T_t^\alpha| \le Me^{\omega t}$ uniformly in $\alpha$ and since $T_t^\alpha$ converges strongly to $T_t$ for each fixed $(t)$ (as argued via resolvent convergence and Laplace inversion), it follows that $|T_t| \le Me^{\omega t}$ as well. Strong continuity of $t \mapsto T_t f$ comes from the dominated convergence in the representation by Yosida approximants and the fact that each $T_t^\alpha$ is strongly continuous; explicitly, for any $f \in \mathcal{H}$,

$$|T_t f - f| \le |T_t f - T_t^\alpha f| + |T_t^\alpha f - f| \qquad (9.31)$$

and the first term tends to 0 as $\alpha \to \infty$ for fixed $t$ (strong convergence), while the second tends to 0 as $t \downarrow 0$ (continuity of the uniformly continuous semigroup $T_t^\alpha$), which yields the desired strong continuity of $T_t$.

Finally, by standard uniqueness theorems for generators of $C_0$-semigroups, the generator of $(T_t)$ is the closure of $L$ (which we assumed closed) and thus equals $L$. This completes the proof of sufficiency.

Combining the two directions yields the equivalence: $L$ is the generator of a strongly continuous semigroup with bound $|T_t| \le Me^{\omega t}$ if and only if $L$ is closed, densely defined, and its resolvent satisfies the uniform bounds

$$| (\alpha I - L)^{-n} | \le M / (\alpha - \omega)^n \qquad (9.32)$$

for all integers $n \ge 1$ and all $\alpha > \omega$.

<figure><img src="image_5.png" /></figure>

In practice one uses the equivalent resolvent estimate for $n = 1$ together with density and closedness. The Hille-Yosida theorem also yields the representation of the resolvent via the
Laplace transform:

$$ (\alpha I - L)^{-1} f = \int_{0}^{\infty} e^{-\alpha t} T_t f dt, \quad \alpha > \omega, f \in \mathcal{H}, \qquad (9.33) $$

which is the basic bridge between the semigroup and generator viewpoints.

Another characterization, tailor-made for contractive semigroups, is given by the Lumer–Phillips theorem.

**Theorem 9.2 (Lumer–Phillips).** A densely defined linear operator $L$ on a Hilbert space $\mathcal{H}$ generates a contraction semigroup $(T_t)_{t \ge 0}$ (i.e. $\|T_t\| \le 1$ for all $t \ge 0$) if and only if $L$ is dissipative

$$ \mathrm{Re}\langle Lf, f \rangle_{\mathcal{H}} \le 0, \quad \forall f \in \mathcal{D}(L), \qquad (9.34) $$

and the range condition $\mathrm{Range}(\lambda I - L) = \mathcal{H}$ holds for some (equivalently all) $\lambda > 0$.

*Proof.* We prove both implications.

(Necessity.) Assume $L$ generates a contraction $C_0$-semigroup $(T_t)_{t \ge 0}$ on $\mathcal{H}$, so $\|T_t\| \le 1$ for all $t \ge 0$. The generator $L$ is densely defined and closed. For any $f \in \mathcal{D}(L)$ consider the function $\phi(t) := \|T_t f\|^2$. Since $t \mapsto T_t f$ is differentiable in norm for $f \in \mathcal{D}(L)$ with derivative $T_t L f$, we compute

$$ \phi'(t) = 2 \mathrm{Re}\langle T_t L f, T_t f \rangle. \qquad (9.35) $$

Contractivity implies $\phi(t) \le \phi(0) = \|f\|^2$ for all $t \ge 0$, hence $\phi'(0) \le 0$. Evaluating at $t = 0$ gives

$$ \phi'(0) = 2 \mathrm{Re}\langle Lf, f \rangle \le 0, \qquad (9.36) $$

which yields the dissipativity condition

$$ \mathrm{Re}\langle Lf, f \rangle \le 0, \quad \forall f \in \mathcal{D}(L). \qquad (9.37) $$

Next, fix $\lambda > 0$. The Laplace transform representation of the resolvent for generators gives

$$ (\lambda I - L)^{-1} f = \int_{0}^{\infty} e^{-\lambda t} T_t f dt, \qquad (9.38) $$

valid for all $f \in \mathcal{H}$. In particular the range of $\lambda I - L$ is all of $\mathcal{H}$, since the resolvent operator is everywhere defined and bounded. Thus the range condition holds for every $\lambda > 0$.

(Sufficiency.) Assume $L$ is densely defined, dissipative, and that $\mathrm{Range}(\lambda I - L) = \mathcal{H}$ for some $\lambda > 0$. We will show that $L$ generates a contraction semigroup.

*Step 1: resolvent estimates.* Fix $\lambda > 0$ with $\mathrm{Range}(\lambda I - L) = \mathcal{H}$. For arbitrary $f \in \mathcal{H}$ let $u \in \mathcal{D}(L)$ be the (unique) solution of $(\lambda I - L)u = f$. Taking the inner product of the equation with $u$ and taking real parts yields

$$ \lambda \|u\|^2 - \mathrm{Re}\langle Lu, u \rangle = \mathrm{Re}\langle f, u \rangle. \qquad (9.39) $$
Using dissipativity Re⟨Lu, u⟩ ≤ 0 we obtain

$$
\lambda \|u\|^2 \le \|f\| \|u\|, \tag{9.40}
$$

hence ‖ u ‖ ≤ ‖ f ‖ /λ . Therefore the resolvent satisfies

$$
\|(\lambda I - L)^{-1}\| \le \frac{1}{\lambda}. \tag{9.41}
$$

In particular the resolvent is a bounded operator on H .

Step 2: Yosida approximants and approximating semigroups. For each $n \in \mathbb{N}$ set $\lambda_n := \lambda + n$ and define the bounded operator

$$
L_n := \lambda_n L (\lambda_n I - L)^{-1} = \lambda_n (I - \lambda_n (\lambda_n I - L)^{-1}). \quad (9.42)
$$

Each $L_n$ is everywhere defined and bounded on $\mathcal{H}$. Moreover for $u \in \mathcal{D}(L)$ one verifies

$$
L_n u = \lambda_n (\lambda_n I - L)^{-1} L u \to Lu \quad (n \to \infty), \quad (9.43)
$$

so $L_n$ converges strongly to $L$ on $\mathcal{D}(L)$. Each $L_n$ generates a uniformly continuous semigroup $e^{tL_n}$ and, using the resolvent estimate from Step 1, one checks the contraction bound

$$
\|e^{tL_n}\| \le 1 \quad \forall t \ge 0, \forall n \in \mathbb{N}. \tag{9.44}
$$

Indeed, for bounded generators one may use the exponential formula and the resolvent bound to obtain contractivity; alternatively one can show dissipativity of $L_n$ and apply the Lumer–Phillips criterion for bounded operators.

Step 3: strong convergence to a contraction semigroup. Fix $f \in \mathcal{H}$. For each $t \ge 0$ the family $(e^{tL_n}f)_{n\in\mathbb{N}}$ is bounded in $\mathcal{H}$. We claim that $e^{tL_n}f$ converges as $n \to \infty$ and that the limit defines a strongly continuous contraction semigroup. To see this, observe that for $\lambda' > \lambda$ the resolvents satisfy

$$
(\lambda' I - L_n)^{-1} = (\lambda' I - L)(\lambda_n I - L)^{-1} (\lambda_n I - L_n)^{-1}, \quad (9.45)
$$

and using the resolvent bounds one establishes that $(\lambda'I-L_n)^{-1}$ converges strongly to $(\lambda'I-L)^{-1}$
as $n \to \infty$. The Laplace inversion formula (Post-Widder) then implies that $e^{tL_n}f$ converges
strongly for each fixed $t$ to some vector $T_t f$, and the limit operators $T_t$ satisfy

$$
(\lambda' I - L)^{-1} = \int_{0}^{\infty} e^{-\lambda' t} T_t dt, \quad \lambda' > \lambda. \tag{9.46}
$$

Using dominated convergence and the uniform contractive bound of $e^{tL_n}$ we deduce $\|T_t\| \le 1$ for every $t \ge 0$. The semigroup property $T_{t+s} = T_t T_s$ follows from taking limits of the semigroup property for $e^{tL_n}$, and strong continuity at $t = 0$ follows from the same limit argument combined with uniform boundedness and the fact that each approximating semigroup is uniformly continuous.
**Step 4: identification of the generator.** It remains to show that the generator of $(T_t)$ is the closure of $L$, which by assumption equals $L$ since $L$ is closed. From the Laplace representation and standard resolvent–generator inversion (compare the Hille–Yosida correspondence) we obtain that the resolvent of the generator of $(T_t)$ equals $(\lambda I - L)^{-1}$ for $\lambda > \lambda_0$, hence the generator coincides with $L$. Therefore $L$ generates the contraction semigroup $(T_t)_{t \ge 0}$.

Combining necessity and sufficiency completes the proof of the theorem. □

In the semigroup-to-generator direction one obtains the dissipativity automatically: if $\|T_t\| \le 1$ then for $f \in \mathcal{D}(L)$,

$$ \mathrm{Re}\langle Lf, f \rangle = \lim_{t \downarrow 0} \frac{\mathrm{Re}\langle T_t f - f, f \rangle}{t} \le 0, \qquad (9.47) $$

since $\|T_t f\| \le \|f\|$. Conversely, dissipativity plus the range condition yields generation of a contraction semigroup.

Important operational tools associated to generators are the Yosida approximants and the resolvent identities. For $\alpha > \omega$ define the bounded Yosida approximation

$$ L_{\alpha} := \alpha L(\alpha I - L)^{-1} = \alpha(I - \alpha(\alpha I - L)^{-1}), \qquad (9.48) $$

which satisfies $L_{\alpha} \in \mathcal{B}(\mathcal{H})$, $\|L_{\alpha}\| \le C(\alpha)$, and for every $f \in \mathcal{D}(L)$,

$$ \lim_{\alpha \to \infty} L_{\alpha} f = Lf. \qquad (9.49) $$

The resolvent identity

$$ (\alpha I - L)^{-1} - (\beta I - L)^{-1} = (\beta - \alpha)(\alpha I - L)^{-1}(\beta I - L)^{-1} \qquad (9.50) $$

is frequently used to pass between different resolvent parameters and to prove strong continuity properties.

We now record several finer properties of the semigroup and generator which are used repeatedly in analysis of PDEs and stochastic processes.

**Domain invariance and regularity.** If $f \in \mathcal{D}(L)$ then $T_t f \in \mathcal{D}(L)$ for all $t \ge 0$ and the map $t \mapsto T_t f$ is $C^1$ with derivative $LT_t f$. Moreover, if $T_t$ is analytic (which occurs, for instance, when $L$ is sectorial), then $T_t$ maps $\mathcal{H}$ into $\mathcal{D}(L^k)$ for every $k \in \mathbb{N}$ and each fixed $t > 0$.

**Approximation by bounded operators (Chernoff product formula).** If $A$ is a bounded operator approximating $I + tL$ at small times in a suitable sense, then

$$ T_t = \lim_{n \to \infty} \left( A\left(\frac{t}{n}\right) \right)^n, \qquad (9.51) $$

with strong convergence on $\mathcal{H}$. The Trotter–Kato product formula is a central instance used to combine semigroups generated by simpler operators to produce the semigroup of a sum.
**Spectral mapping and growth/decay rates.** The spectral mapping theorem for $C_0$-semigroups states that for the point spectrum,

$$e^{t\sigma_p(L)} \subset \sigma_p(T_t), \qquad (9.52)$$

and for the full spectrum one has the inclusion $e^{t\sigma_p(L)} \setminus \{0\} \subset \sigma(T_t)$; equality may fail in full generality but holds under additional hypotheses (e.g. boundedness or normality). The growth bound $\omega_0$ defined by

$$\omega_0 := \inf\{\omega \in \mathbb{R} : \exists M \ge 1, \|T_t\| \le Me^{\omega t} \forall t \ge 0\} \qquad (9.53)$$

satisfies $\omega_0 \ge s(L) := \sup\{\text{Re} \lambda : \lambda \in \sigma(L)\}$ and in many self-adjoint or sectorial settings equality holds.

**Cores and perturbation stability.** A subspace $C \subset D(L)$ is called a core for $L$ if the closure of $L|_C$ equals $L$. In practice one often verifies generation or properties of $L$ on a convenient core (e.g. $C_c^\infty$ for differential generators) and extends by closure. Under relatively bounded perturbations (Kato–Rellich type), generation and essential properties of the semigroup persist; specifically, if $B$ is $L$-bounded with relative bound $a < 1$ then $L+B$ is closed and generates a $C_0$-semigroup.

**Concrete examples.** The prototypical examples on $L^2(\mathbb{R}^d)$ are the (scaled) Laplacian $(\frac{1}{2}\Delta, H^2(\mathbb{R}^d))$ generating the heat semigroup $T_t f = p_t * f$, and second-order uniformly elliptic operators with lower-order drift terms, which under suitable form or sectorial conditions generate analytic semigroups. In probability these are the transition semigroups of diffusion processes and form the bridge to Dirichlet-form constructions.

In summary, strongly continuous semigroups on Hilbert spaces are completely characterized by their infinitesimal generators via Hille–Yosida and Lumer–Phillips theorems; resolvent and Yosida approximations provide constructive tools; cores and perturbation results enable verification in concrete settings; and spectral mapping plus analyticity furnish fine regularity and long-time asymptotics essential for both PDE and stochastic analyses.

## 9.2 Self-Adjoint Operators and Spectral Theorem

Let $(T_t)_{t \ge 0}$ be a strongly continuous semigroup on the Hilbert space $L^2(E, \mu)$ such that for all $f, g \in L^2(E, \mu)$ and all $t \ge 0$,

$$\langle T_t f, g \rangle_{L^2(E, \mu)} = \langle f, T_t g \rangle_{L^2(E, \mu)}. \qquad (9.54)$$

That is, each $T_t$ is self-adjoint. We will rigorously show that its infinitesimal generator $L$ is self-adjoint, nonpositive, and that the spectral theorem provides a complete functional calculus for $L$ and $(T_t)$.
## 9.2.1 Self-adjointness of the generator

**Lemma 9.3.** If $(T_t)_{t \ge 0}$ is a symmetric, strongly continuous semigroup of contractions on $L^2(E, \mu)$, then its infinitesimal generator $L$ is self-adjoint and satisfies $\langle Lf, f \rangle_{L^2(E, \mu)} \le 0$ for all $f \in \mathcal{D}(L)$.

*Proof.* Let $f, g \in \mathcal{D}(L)$ and recall that by definition of the generator,

$$L f = \lim_{t \downarrow 0} \frac{T_t f - f}{t}, \quad \text{in } L^2(E, \mu). \qquad (9.55)$$

Since $T_t$ is symmetric, we have for all $t > 0$,

$$\langle T_t f, g \rangle = \langle f, T_t g \rangle. \qquad (9.56)$$

Dividing both sides by $t$ and subtracting $\langle f, g \rangle / t$ yields

$$\left\langle \frac{T_t f - f}{t}, g \right\rangle = \left\langle f, \frac{T_t g - g}{t} \right\rangle. \qquad (9.57)$$

Passing to the limit $t \downarrow 0$ gives

$$\langle Lf, g \rangle = \langle f, Lg \rangle, \quad \forall f, g \in \mathcal{D}(L), \qquad (9.58)$$

showing that $L$ is symmetric. Furthermore, contractivity implies $\|T_t f\| \le \|f\|$, and differentiating $\|T_t f\|^2$ at $t = 0$ yields

$$\left. \frac{d}{dt} \|T_t f\|^2 \right|_{t=0} = 2 \operatorname{Re} \langle Lf, f \rangle \le 0, \qquad (9.59)$$

so $\operatorname{Re}\langle Lf, f \rangle \le 0$. Thus $L$ is dissipative and symmetric. By a standard result in semigroup theory (see, e.g., Reed-Simon, Functional Analysis, Thm. X.49), a densely defined symmetric dissipative operator generating a contraction semigroup is self-adjoint and nonpositive. $\square$

## 9.2.2 Spectral representation

Since $L$ is self-adjoint and nonpositive, the spectral theorem ensures the existence of a unique projection-valued measure $E_\lambda$ on $\mathbb{R}_-$ such that

$$L = \int_{\mathbb{R}_-} \lambda dE_{\lambda}. \qquad (9.60)$$

For any Borel measurable function $\varphi : \mathbb{R}_- \to \mathbb{C}$, we define

$$\varphi(L) = \int_{\mathbb{R}_-} \varphi(\lambda) dE_{\lambda}, \qquad (9.61)$$

which gives the full functional calculus for $L$. In particular, setting $\varphi_t(\lambda) = e^{t\lambda}$, we obtain the semigroup representation

$$T_t = e^{tL} = \int_{\mathbb{R}_-} e^{t\lambda} dE_{\lambda}, \quad t \ge 0. \qquad (9.62)$$
This representation is strongly continuous in $t$, and since $|e^{t\lambda}| \le 1$ for $\lambda \le 0$, we have $\|T_t\| \le 1$, confirming that $(T_t)$ is a contraction semigroup. Moreover, differentiability in $t$ follows from dominated convergence under the spectral integral, hence $(T_t)$ is analytic.

### 9.2.3 Quadratic form and fractional powers

Define the quadratic form $\mathcal{E}$ associated with $L$ by

$$ \mathcal{E}(f, g) := -\langle Lf, g \rangle, \quad f, g \in \mathcal{D}(L). \qquad (9.63) $$

Since $L$ is self-adjoint and nonpositive, $\mathcal{E}$ is symmetric, closed, and positive definite. The spectral calculus allows us to define fractional powers of $(-L)$ by

$$ (-L)^{\alpha} = \int_{\mathbb{R}_{-}} (-\lambda)^{\alpha} dE_{\lambda}, \quad \alpha > 0. \qquad (9.64) $$

In particular, the domain of the square root is given by

$$ \mathcal{D}((-L)^{1/2}) = \left\{ f \in L^2(E, \mu) : \int_{\mathbb{R}_{-}} (-\lambda) d\langle E_{\lambda}f, f \rangle < \infty \right\}, \qquad (9.65) $$

and satisfies

$$ \|(-L)^{1/2} f\|_{2}^{2} = \int_{\mathbb{R}_{-}} (-\lambda) d\langle E_{\lambda}f, f \rangle = \mathcal{E}(f, f). \qquad (9.66) $$

Hence the quadratic form $\mathcal{E}$ coincides with the Dirichlet form associated to $L$, and the correspondence

$$ \mathcal{E}(f, f) = \|(-L)^{1/2} f\|_{2}^{2}, \quad \mathcal{D}(\mathcal{E}) = \mathcal{D}((-L)^{1/2}), \qquad (9.67) $$

establishes the equivalence between the analytic structure defined by the form and the spectral properties of the self-adjoint generator.

## 9.3 Compactness and Spectral Discreteness

A central analytic property in the study of self-adjoint generators of Dirichlet forms and Schrödinger-type operators concerns the compactness of the semigroup $(T_t)_{t \ge 0}$. Compactness has deep implications for the spectral structure of the generator $L$ and the asymptotic behaviour of the semigroup.

### 9.3.1 Compactness criterion and its consequences

Let $(T_t)_{t \ge 0}$ be a strongly continuous, symmetric semigroup on the Hilbert space $L^2(E, \mu)$ with self-adjoint, nonpositive generator $L$. We say that $T_t$ is compact if, for some (hence all) $t > 0$, the operator $T_t : L^2(E, \mu) \to L^2(E, \mu)$ is compact, i.e. it maps bounded sets into relatively compact sets.

**Lemma 9.4.** If $T_t$ is compact for one $t > 0$, then $T_s$ is compact for all $s > 0$. Moreover, $L$ has purely discrete spectrum.
Proof. Suppose that $T_{t_0}$ is compact for some $t_0 > 0$. For any $s > 0$, write

$$ T_s = \begin{cases} T_{t_0}^{\lfloor s/t_0 \rfloor} T_r, & \text{if } s > t_0, \text{ with } r = s - \lfloor s/t_0 \rfloor t_0, \\ T_{t_0}^{1/2} T_{s-t_0/2} T_{t_0}^{1/2}, & \text{if } s < t_0. \end{cases} \qquad (9.68) $$

Since products of compact and bounded operators are compact, it follows that $T_s$ is compact for every $s > 0$. Hence, $T_t$ forms a compact semigroup.

By the spectral theorem, the compactness of $T_t$ implies that its spectrum $\sigma(T_t)$ consists of a countable sequence of real eigenvalues $\{e^{-t\lambda_n}\}_{n=1}^{\infty}$ accumulating only at 0, each with finite multiplicity. Correspondingly, the spectrum of the generator $L$ is given by

$$ \sigma(L) = \{-\lambda_n\}_{n=1}^{\infty}, \quad \lambda_1 < \lambda_2 \le \lambda_3 \le \dots, \quad \lambda_n \to +\infty. \qquad (9.69) $$

Each $\lambda_n > 0$ is an eigenvalue of $-L$ with eigenfunction $\phi_n$, satisfying

$$ L\phi_n = -\lambda_n\phi_n. \qquad (9.70) $$

Since $T_t = e^{tL}$, we have

$$ T_t \phi_n = e^{-t\lambda_n} \phi_n. \qquad (9.71) $$

The family $\{\phi_n\}_{n \ge 1}$ forms an orthonormal basis of $L^2(E, \mu)$, by the completeness property of spectral projections associated with compact self-adjoint operators. $\square$

### 9.3.2 Examples and sufficient conditions for compactness

Compactness of $(T_t)$ (and hence discreteness of the spectrum of $L$) arises in several important contexts:

(i) **Finite measure space:** If $\mu(E) < \infty$ and $(T_t)$ is a symmetric, positivity-preserving Markov semigroup (e.g. the heat semigroup on a bounded domain with Dirichlet boundary conditions), then compactness of $T_t$ follows from the Rellich-Kondrachov compact embedding theorem:

$$ H_0^1(E) \hookrightarrow L^2(E, \mu) \text{ compactly,} \qquad (9.72) $$

which ensures that the corresponding generator (typically $-\frac{1}{2}\Delta$) has purely discrete spectrum.

(ii) **Confining potential:** For the Schrödinger operator

$$ H = -\frac{1}{2}\Delta + V(x) \qquad (9.73) $$

on $\mathbb{R}^d$, if the potential $V(x)$ satisfies $V(x) \to +\infty$ as $|x| \to \infty$, then the embedding of the associated form domain

$$ \mathcal{D}(\mathcal{E}^V) = \{f \in H^1(\mathbb{R}^d) : V^{1/2}f \in L^2(\mathbb{R}^d)\} \qquad (9.74) $$

into $L^2(\mathbb{R}^d)$ is compact. Consequently, $e^{-tH}$ is a compact operator for all $t > 0$, and
the spectrum of $H$ consists of discrete eigenvalues $\{\lambda_n\}_{n=1}^\infty$ with finite multiplicities and $\lambda_n \to +\infty$.

(iii) **Elliptic operators on compact manifolds:** If $E$ is a compact Riemannian manifold and $L$ is the Laplace-Beltrami operator, then $T_t = e^{tL}$ is compact on $L^2(E)$ because the embedding $H^1(E) \hookrightarrow L^2(E)$ is compact. In this case, the eigenfunctions $\{\phi_n\}$ form an orthonormal basis of smooth functions on $E$, and the eigenvalues $\{\lambda_n\}$ satisfy Weyl's asymptotic law:

$$ \lambda_n \sim C_d n^{2/d}, \quad n \to \infty. \qquad (9.75) $$

### 9.3.3 Summary of implications

Hence, the following equivalence holds for a symmetric semigroup $(T_t)$ with self-adjoint generator $L$:

$$ T_t \text{ compact for some } t > 0 \iff L \text{ has purely discrete spectrum.} \qquad (9.76) $$

In that case, we have the spectral decompositions

$$ T_t f = \sum_{n=1}^{\infty} e^{-t\lambda_n} \langle f, \phi_n \rangle \phi_n, \qquad (9.77) $$

$$ Lf = - \sum_{n=1}^{\infty} \lambda_n \langle f, \phi_n \rangle \phi_n, \qquad (9.78) $$

for all $f \in L^2(E, \mu)$, with $\{\phi_n\}$ forming a complete orthonormal basis and $\lambda_n \to +\infty$.

## 9.4 Form Methods and the Friedrichs Extension

Let $\mathcal{H} = L^2(E, \mu)$ be a separable Hilbert space, and let $\mathcal{E} : \mathcal{D}(\mathcal{E}) \times \mathcal{D}(\mathcal{E}) \to \mathbb{R}$ be a densely defined, symmetric bilinear form satisfying

$$ \mathcal{E}(f, g) = \mathcal{E}(g, f), \quad \forall f, g \in \mathcal{D}(\mathcal{E}). \qquad (9.79) $$

Assume moreover that $\mathcal{E}$ is nonnegative, i.e.

$$ \mathcal{E}(f, f) \ge 0, \quad \forall f \in \mathcal{D}(\mathcal{E}). \qquad (9.80) $$

**Closability of Forms:** A form $\mathcal{E}$ is said to be *closable* if for every sequence $(f_n) \subset \mathcal{D}(\mathcal{E})$ satisfying

$$ f_n \to 0 \text{ in } L^2(E, \mu), \quad \text{and} \quad \mathcal{E}(f_n - f_m, f_n - f_m) \to 0, $$

we have

$$ \mathcal{E}(f_n, f_n) \to 0. $$

In this case, there exists a smallest closed extension $(\bar{\mathcal{E}}, \mathcal{D}(\bar{\mathcal{E}}))$ such that $\mathcal{E} \subset \bar{\mathcal{E}}$, obtained by completing $\mathcal{D}(\mathcal{E})$ under the norm

$$ \|f\|_{\mathcal{E}} := (\mathcal{E}(f, f) + \|f\|_{2}^{2})^{1/2}. \qquad (9.81) $$

The pair $(\bar{\mathcal{E}}, \mathcal{D}(\bar{\mathcal{E}}))$ is called a closed symmetric form on $L^2(E, \mu)$.
9.4.1 First Representation Theorem (Kato–Friedrichs Theorem)

A central result connecting closed forms and self-adjoint operators is the first representation theorem. Let ($\mathcal{E}$, $\mathcal{D}(\mathcal{E})$) be a densely defined, symmetric, closed, nonnegative form on $\mathcal{H}$. Then there exists a unique nonnegative self-adjoint operator $L$ on $\mathcal{H}$ such that

$$ \mathcal{D}(\mathcal{E}) = \mathcal{D}((-L)^{1/2}), \quad \mathcal{E}(f, g) = \langle (-L)^{1/2} f, (-L)^{1/2} g \rangle_{L^2}, \quad \forall f, g \in \mathcal{D}(\mathcal{E}). \quad (9.82) $$

Conversely, given a nonnegative self-adjoint operator $L$, the bilinear form defined by the right-hand side above is closed, symmetric, and nonnegative.

*Proof.* Let ($\mathcal{E}$, $\mathcal{D}(\mathcal{E})$) be a densely defined, symmetric, closed, nonnegative form on the Hilbert space $\mathcal{H}$. Define the energy norm $\| \cdot \|_{\mathcal{E}}$ on $\mathcal{D}(\mathcal{E})$ by

$$ \|u\|_{\mathcal{E}} := (\mathcal{E}(u, u) + \|u\|_{\mathcal{H}}^2)^{1/2}. \quad (9.83) $$

Closedness of $\mathcal{E}$ means that ($\mathcal{D}(\mathcal{E})$, $\| \cdot \|_{\mathcal{E}}$) is a Hilbert space. The canonical embedding $j : (\mathcal{D}(\mathcal{E}), \| \cdot \|_{\mathcal{E}}) \hookrightarrow \mathcal{H}$ is continuous and has dense range because $\mathcal{D}(\mathcal{E})$ is dense in $\mathcal{H}$.

For each fixed $u \in \mathcal{D}(\mathcal{E})$ the map $\mathcal{D}(\mathcal{E}) \ni v \mapsto \mathcal{E}(u, v)$ is a bounded linear functional on $(\mathcal{D}(\mathcal{E}), \| \cdot \|_{\mathcal{E}})$. Indeed, by the Cauchy–Schwarz inequality for the form,

$$ |\mathcal{E}(u, v)| \le \mathcal{E}(u, u)^{1/2} \mathcal{E}(v, v)^{1/2} \le \mathcal{E}(u, u)^{1/2} \|v\|_{\mathcal{E}}. \quad (9.84) $$

Hence, by the Riesz representation theorem in the Hilbert space ($\mathcal{D}(\mathcal{E})$, $\| \cdot \|_{\mathcal{E}}$), there exists a unique element $\tilde{Lu} \in \mathcal{D}(\mathcal{E})$ (viewed as the Riesz representative with respect to $\langle \cdot, \cdot \rangle_{\mathcal{E}}$) such that for all $v \in \mathcal{D}(\mathcal{E})$,

$$ \mathcal{E}(u, v) + \langle u, v \rangle_{\mathcal{H}} = \langle \tilde{L}u, v \rangle_{\mathcal{E}}. \quad (9.85) $$

Unwinding the right-hand side in terms of the energy inner product yields, by definition of $\langle \cdot, \cdot \rangle_{\mathcal{E}}$,

$$ \mathcal{E}(u, v) + \langle u, v \rangle_{\mathcal{H}} = \mathcal{E}(\tilde{L}u, v) + \langle \tilde{L}u, v \rangle_{\mathcal{H}}. \quad (9.86) $$

Rearranging gives the symmetric identity

$$ \mathcal{E}(u - \tilde{L}u, v) + \langle u - \tilde{L}u, v \rangle_{\mathcal{H}} = 0, \quad \forall v \in \mathcal{D}(\mathcal{E}). \quad (9.87) $$

Since the left-hand side is the $\mathcal{E}$-inner product with $u - \tilde{L}u$, this implies $u = \tilde{L}u$ as an identity in the Hilbert space ($\mathcal{D}(\mathcal{E})$, $\| \cdot \|_{\mathcal{E}}$). Thus the Riesz representative is precisely $u$ itself and the previous representation can be rewritten to exhibit a linear operator relating the form to the Hilbert-space inner product.

Define the linear operator $A : \mathcal{D}(A) \subset \mathcal{H} \to \mathcal{H}$ by declaring $u \in \mathcal{D}(A)$ if and only if the linear functional $\mathcal{D}(\mathcal{E}) \ni v \mapsto \mathcal{E}(u, v)$ is bounded in the $\mathcal{H}$-norm, i.e. there exists $h \in \mathcal{H}$ such that

$$ \mathcal{E}(u, v) = \langle h, v \rangle_{\mathcal{H}}, \quad \forall v \in \mathcal{D}(\mathcal{E}). \quad (9.88) $$
For such $u$ we set $Au := h$. Equivalently,

$$ \mathcal{D}(A) = \{u \in \mathcal{D}(\mathcal{E}) : \exists h \in \mathcal{H} \text{ s.t. } \mathcal{E}(u, v) = \langle h, v \rangle_{\mathcal{H}} \quad \forall v \in \mathcal{D}(\mathcal{E})\}, \quad (9.89) $$

and $A : \mathcal{D}(A) \to \mathcal{H}$ is linear by construction. Observe that $\mathcal{D}(A)$ is dense in $\mathcal{H}$ because $\mathcal{D}(\mathcal{E})$ is dense and the form is closed.

We now show that $A$ is self-adjoint and nonnegative. First, for $u \in \mathcal{D}(A)$ and $v \in \mathcal{D}(\mathcal{E})$ we have

$$ \langle Au, v \rangle_{\mathcal{H}} = \mathcal{E}(u, v) = \mathcal{E}(v, u) = \overline{\langle Av, u \rangle_{\mathcal{H}}}, \quad (9.90) $$

whenever $v \in \mathcal{D}(A)$ as well; hence $A$ is symmetric on $\mathcal{D}(A)$. Next, for $u \in \mathcal{D}(A)$ taking $v = u$ in the defining relation gives

$$ \langle Au, u \rangle_{\mathcal{H}} = \mathcal{E}(u, u) \geq 0, \quad (9.91) $$

so $A$ is nonnegative.

To see that $A$ is self-adjoint, consider the operator $B := I + A$ with domain $\mathcal{D}(B) = \mathcal{D}(A)$. For $u \in \mathcal{D}(B)$ and $v \in \mathcal{D}(\mathcal{E})$ we have

$$ \langle Bu, v \rangle_{\mathcal{H}} = \langle u, v \rangle_{\mathcal{H}} + \mathcal{E}(u, v) =: \ell_u(v), \quad (9.92) $$

and the map $v \mapsto \ell_u(v)$ is a bounded linear functional on $(\mathcal{D}(\mathcal{E}), \| \cdot \|_{\mathcal{E}})$. By the Riesz representation theorem there exists a unique $w \in \mathcal{D}(\mathcal{E})$ such that $\ell_u(v) = \langle w, v \rangle_{\mathcal{E}}$ for all $v$. Unwinding definitions shows $B$ is surjective onto $\mathcal{H}$ and has bounded inverse $B^{-1} : \mathcal{H} \to \mathcal{D}(\mathcal{E}) \subset \mathcal{H}$. In particular $B$ is self-adjoint and strictly positive, hence $A = B - I$ is self-adjoint and nonnegative. We therefore set $L := A$, obtaining a self-adjoint nonnegative operator associated with the form.

It remains to identify the form domain with the domain of the square root of $-L$. Since $L$ is nonnegative and self-adjoint, functional calculus yields the square root $(-L)^{1/2}$ and its domain $\mathcal{D}((-L)^{1/2})$. For $u \in \mathcal{D}(A)$ we have

$$ \|(-L)^{1/2} u\|_{\mathcal{H}}^2 = \langle -Lu, u \rangle_{\mathcal{H}} = \mathcal{E}(u, u), \quad (9.93) $$

so $\mathcal{D}(A) \subset \mathcal{D}((-L)^{1/2})$ and the equality of norms shows that the form $\mathcal{E}$ coincides with the quadratic form induced by $(-L)^{1/2}$ on $\mathcal{D}(A)$. By closedness and density the identity extends to

$$ \mathcal{D}(\mathcal{E}) = \mathcal{D}((-L)^{1/2}), \quad \mathcal{E}(u, v) = \langle (-L)^{1/2} u, (-L)^{1/2} v \rangle_{\mathcal{H}}, \quad \forall u, v \in \mathcal{D}(\mathcal{E}). \quad (9.94) $$

Uniqueness of $L$ follows because any self-adjoint nonnegative operator $L'$ satisfying the displayed identity must have the same quadratic form and hence the same spectral measure; by functional calculus the operators $L$ and $L'$ coincide.

Conversely, if $L$ is a given nonnegative self-adjoint operator on $\mathcal{H}$, define the quadratic form $\mathcal{E}_L$ by

$$ \mathcal{D}(\mathcal{E}_L) := \mathcal{D}((-L)^{1/2}), \quad \mathcal{E}_L(u, v) := \langle (-L)^{1/2} u, (-L)^{1/2} v \rangle_{\mathcal{H}}. \quad (9.95) $$

By the spectral theorem this form is densely defined, symmetric, closed and nonnegative, and
the correspondence constructed above is inverted. This completes the proof of the first representation theorem. □

## 9.4.2 Lower Bounded Forms and Friedrichs Extension

If $\mathcal{E}$ is not nonnegative but *lower bounded*, i.e.

$$ \mathcal{E}(f, f) \geq -C\|f\|_2^2, \quad \text{for some } C \in \mathbb{R}, \qquad (9.96) $$

then we can modify the form to make it nonnegative by setting

$$ \mathcal{E}_C(f, g) := \mathcal{E}(f, g) + C\langle f, g \rangle. \qquad (9.97) $$

The closedness of $\mathcal{E}_C$ implies that $\mathcal{E}$ is also closed in the sense of lower semibounded forms. In this case, the first representation theorem still applies and produces a unique self-adjoint operator $L$ such that

$$ \mathcal{E}(f, g) = \langle (-L)^{1/2} f, (-L)^{1/2} g \rangle - C \langle f, g \rangle. \qquad (9.98) $$

If $\mathcal{E}$ is the restriction of a symmetric differential operator $A$ defined on a dense subspace $D_0 \subset L^2(E, \mu)$ (e.g. $C_c^\infty(E)$), then the self-adjoint operator associated with the closure of $\mathcal{E}$ is called the *Friedrichs extension* of $A$. Formally, if $A$ is symmetric and lower bounded on $D_0$, the Friedrichs extension $A_F$ is the unique self-adjoint operator satisfying

$$ D(A_F^{1/2}) = \overline{D_0}^{\|\cdot\|_{\mathcal{E}}}, \quad \mathcal{E}(f, g) = \langle A_F^{1/2} f, A_F^{1/2} g \rangle - C \langle f, g \rangle. \qquad (9.99) $$

## 9.4.3 Connection with Dirichlet Forms

In the context of diffusion processes, the symmetric closed form $\mathcal{E}$ represents the *Dirichlet form* associated to the infinitesimal generator of the process. The Friedrichs extension ensures the existence of a self-adjoint realization of the (possibly unbounded) operator describing the energy dissipation, even when the original domain is only dense and not closed. This framework provides a rigorous bridge between partial differential operators, spectral theory, and stochastic analysis.

# 9.5 Analyticity, Sectorial Forms, and Holomorphic Semigroups

Let $\mathcal{H} = L^2(E, \mu)$ be a complex Hilbert space. We consider a sesquilinear form $\mathcal{E} : \mathcal{D}(\mathcal{E}) \times \mathcal{D}(\mathcal{E}) \to \mathbb{C}$ that is densely defined, closed, and *sectorial*. That is, there exist constants $\theta \in [0, \frac{\pi}{2})$ and $C > 0$ such that for all $f, g \in \mathcal{D}(\mathcal{E})$:

$$ |\operatorname{Im} \mathcal{E}(f, f)| \leq \tan(\theta) \operatorname{Re} \mathcal{E}(f, f), \quad \text{and} \quad |\mathcal{E}(f, g)| \leq C [\operatorname{Re} \mathcal{E}(f, f)]^{1/2} [\operatorname{Re} \mathcal{E}(g, g)]^{1/2}. \qquad (9.100) $$

In this case, the numerical range of $\mathcal{E}$ lies in the closed sector

$$ \Sigma_{\theta} := \{z \in \mathbb{C} : |\arg z| \leq \theta\}, \qquad (9.101) $$

and the form $\mathcal{E}$ is said to be *sectorial with semi-angle $\theta$*.
**Closability and the Associated Operator:** If $\mathcal{E}$ satisfies the sector condition (9.100) and is closable, its closure $(\bar{\mathcal{E}}, \mathcal{D}(\bar{\mathcal{E}}))$ defines a unique $m$-sectorial operator $L$ on $\mathcal{H}$ via the second representation theorem (Kato):

$$ \mathcal{E}(f, g) = \langle Lf, g \rangle, \quad \forall f \in \mathcal{D}(L), \forall g \in \mathcal{D}(\mathcal{E}). \qquad (9.102) $$

Here, $L$ is the maximal accretive (m-accretive) operator associated to $\mathcal{E}$, characterized by

$$ \mathrm{Re}\langle Lf, f \rangle \geq 0, \quad \forall f \in \mathcal{D}(L), \qquad (9.103) $$

and $\lambda I + L$ is surjective for some (hence all) $\lambda > 0$.

**M-Sectorial Operators and Holomorphic Semigroups:** An operator $L$ on $\mathcal{H}$ is said to be $m$-sectorial if it is closed, densely defined, and satisfies:

1. Its numerical range satisfies

$$ W(L) := \{\langle Lf, f \rangle : f \in \mathcal{D}(L), \|f\| = 1\} \subseteq \Sigma_{\theta}, \qquad (9.104) $$

for some $\theta \in [0, \pi/2)$;

2. The range condition holds:

$$ \mathrm{Range}(\lambda I + L) = \mathcal{H}, \quad \forall \lambda > 0. \qquad (9.105) $$

A fundamental theorem of semigroup theory (Kato, 1953) states that if $L$ is $m$-sectorial, then $-L$ generates a strongly continuous holomorphic semigroup $(T_t)_{t \ge 0}$ satisfying

$$ T_t = e^{-tL}, \quad \text{and} \quad T_z = e^{-zL} \text{ is holomorphic in } z \in \Sigma_{\frac{\pi}{2}-\theta}. \qquad (9.106) $$

Thus, the analyticity of $(T_t)$ in a sector of the complex plane is a direct consequence of the sectoriality of $\mathcal{E}$ (or equivalently, $L$).

**Angle of Analyticity and Norm Bounds:** For $L$ $m$-sectorial with semi-angle $\theta < \frac{\pi}{2}$, the generated semigroup $(T_t)$ extends analytically to the sector

$$ S_{\frac{\pi}{2}-\theta} := \{z \in \mathbb{C} : |\arg z| < \frac{\pi}{2} - \theta\}, \qquad (9.107) $$

and satisfies the bound

$$ \|T_z\| \le e^{-\mathrm{Re}(z)\omega}, \quad z \in S_{\frac{\pi}{2}-\theta}, \qquad (9.108) $$

for some $\omega \in \mathbb{R}$ depending on the lower bound of $\mathrm{Re}\mathcal{E}(f, f)$. In particular, $(T_t)$ is analytic on the real half-axis and exponentially bounded.

**Examples (Non-Symmetric Diffusion Operators):** Sectorial forms naturally arise in the study of elliptic operators with drift. For instance, consider on $\mathcal{H} = L^2(\mathbb{R}^d, dx)$ the bilinear form

$$ \mathcal{E}(f, g) = \int_{\mathbb{R}^d} \nabla f(x) \cdot \overline{\nabla g(x)} dx + \int_{\mathbb{R}^d} b(x) \cdot \overline{\nabla f(x) g(x)} dx, \qquad (9.109) $$
with div b = 0 and b ∈ L $^{∞}$( R $^{d}$, R $^{d}$). The second term introduces a non-symmetric component but satisfies the sector condition provided ‖ b ‖ ∞ is sufficiently small relative to the diffusion coefficient. The associated operator

$$
Lf = \Delta f - b \cdot \nabla f \tag{9.110}
$$

is m-sectorial and thus generates a holomorphic semigroup (T_t)_{t≥0} on L²(ℝ^d), corresponding to the non-symmetric diffusion process with drift b.

**Summary:** The sector condition ensures that the imaginary part of $\mathcal{E}(f, f)$ is controlled by its real part, guaranteeing accretivity and analyticity of the generated semigroup. Hence, the passage from symmetric to sectorial forms generalizes the Dirichlet form framework to non-symmetric diffusion operators, allowing one to rigorously treat elliptic operators with drift or complex coefficients within the theory of holomorphic semigroups.

9.6 Spectral Measures, Functional Calculus, and Resolvent Identities

Let $L$ be a self-adjoint operator on the Hilbert space $\mathcal{H} = L^2(E, \mu)$. By the spectral theorem there exists a unique projection-valued measure $E(\cdot)$ defined on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ such that for every bounded Borel measurable function $\varphi : \mathbb{R} \to \mathbb{C}$ the operator

$$
\varphi(L) := \int_{\mathbb{R}} \varphi(\lambda) dE(\lambda) \qquad (9.111)
$$

is a bounded linear operator on $\mathcal{H}$ and the map $\varphi \mapsto \varphi(L)$ is a *-homomorphism from $B_b(\mathbb{R})$ (the bounded Borel functions) into $\mathcal{B}(\mathcal{H})$. Moreover, for each $f \in \mathcal{H}$ the complex measure $\mu_f(A) := \langle E(A)f, f \rangle$ is finite and

$$
\|\varphi(L)f\|^2 = \int_{\mathbb{R}} |\varphi(\lambda)|^2 d\mu_f(\lambda). \quad (9.112)
$$

The spectral representation yields the following important special cases (we record them in the
form convenient for semigroup and resolvent theory). If $\sigma(L)$ denotes the spectrum of $L$, then
for $\alpha \in \mathbb{C} \setminus \sigma(L)$ the resolvent $(\alpha I - L)^{-1}$ exists and admits the spectral integral representation

$$
R(\alpha) := (\alpha I - L)^{-1} = \int_{\sigma(L)} \frac{1}{\alpha - \lambda} dE(\lambda). \quad (9.113)
$$

Since the integrand $\lambda \mapsto (\alpha - \lambda)^{-1}$ is bounded on $\sigma(L)$, the operator on the right-hand side is bounded and equal to the resolvent. In particular, for $\alpha$ in the resolvent set one has the norm bound

$$
\| R(\alpha) \| \le \sup_{\lambda \in \sigma(L)} \frac{1}{|\alpha - \lambda|}. \qquad (9.114)
$$

**Resolvent identity.** For any $\alpha, \beta \in \mathbb{C} \setminus \sigma(L)$ the following algebraic identity holds:

$$
R(\alpha) - R(\beta) = (\beta - \alpha)R(\alpha)R(\beta). \tag{9.115}
$$
Proof: using the spectral representation,

$$ 
\begin{aligned}
R(\alpha) - R(\beta) &= \int_{\sigma(L)} \left( \frac{1}{\alpha - \lambda} - \frac{1}{\beta - \lambda} \right) dE(\lambda) = \int_{\sigma(L)} \frac{\beta - \alpha}{(\alpha - \lambda)(\beta - \lambda)} dE(\lambda) \\
&= (\beta - \alpha) \int_{\sigma(L)} \frac{1}{\alpha - \lambda} \frac{1}{\beta - \lambda} dE(\lambda) = (\beta - \alpha)R(\alpha)R(\beta),
\end{aligned} 
\quad (9.116) $$

where the last equality follows from the multiplicative property of the functional calculus for bounded Borel functions. This identity is the cornerstone of resolvent calculus and yields many classical consequences (e.g. resolvent expansions, perturbation formulae).

**Functional calculus and algebraic properties.** The spectral calculus preserves algebraic operations and adjoints: if $\varphi, \psi \in B_b(\mathbb{R})$ then

$$ \varphi(L)\psi(L) = (\varphi\psi)(L), \quad \varphi(L)^* = \bar{\varphi}(L). \quad (9.117) $$

In particular, exponential and fractional powers are defined via the spectral integral:

$$ e^{tL} = \int_{\sigma(L)} e^{t\lambda} dE(\lambda), \quad (-L)^{\alpha} = \int_{\sigma(L)} (-\lambda)^{\alpha} dE(\lambda), \quad (9.118) $$

whenever the right-hand side is well-defined (for the exponential the integrand is bounded on any spectral subset bounded above).

**Semigroup generation via spectral calculus.** If $L$ is self-adjoint and bounded above by some constant $\omega$ (for instance, when $-L$ is nonnegative), then the family $(T_t)_{t \ge 0}$ defined by

$$ T_t := e^{tL} = \int_{\sigma(L)} e^{t\lambda} dE(\lambda) \quad (9.119) $$

is a strongly continuous one-parameter semigroup with generator $L$. Strong continuity follows from dominated convergence applied to the spectral integral: for any fixed $f \in \mathcal{H}$,

$$ \|T_t f - f\|^2 = \int_{\sigma(L)} |e^{t\lambda} - 1|^2 d\mu_f(\lambda) \to 0 \quad (t \downarrow 0), \quad (9.120) $$

because $|e^{t\lambda} - 1|^2 \le C|t|^2(1 + \lambda^2)$ locally and $\mu_f$ has finite second moment on bounded spectral sets. In the common probabilistic convention where $L$ is the (negative) generator of a contraction semigroup, one usually writes $T_t = e^{tL}$ with $\sigma(L) \subset (-\infty, 0]$, so $T_t$ is contractive.

**Laplace transform and inversion.** The resolvent and semigroup are related by the (operator-valued) Laplace transform. Suppose $L$ is such that $e^{tL}$ is exponentially bounded, i.e. $\|e^{tL}\| \le Me^{\omega t}$. Then for $\alpha > \omega$ the Laplace transform identity holds in the strong operator sense:

$$ (\alpha I - L)^{-1} = \int_{0}^{\infty} e^{-\alpha t} e^{tL} dt, \quad (9.121) $$

where the Bochner integral converges in operator norm on $\mathcal{H}$ due to the exponential bound. The proof is immediate from the spectral representations: for $\alpha$ in the right half-plane,

$$ \int_{0}^{\infty} e^{-\alpha t} e^{tL} dt = \int_{0}^{\infty} e^{-\alpha t} \left( \int_{\sigma(L)} e^{t\lambda} dE(\lambda) \right) dt = \int_{\sigma(L)} \left( \int_{0}^{\infty} e^{-(\alpha-\lambda)t} dt \right) dE(\lambda) = \int_{\sigma(L)} \frac{1}{\alpha - \lambda} dE(\lambda), \quad (9.122) $$
justifying the resolvent representation. Conversely, given the resolvent one recovers the semi-group by the Bromwich (inverse Laplace) transform in the strong operator topology:

$$e^{tL} = \frac{1}{2\pi i} \lim_{R \to \infty} \int_{\gamma_R} e^{t\alpha} (\alpha I - L)^{-1} d\alpha, \qquad (9.123)$$

where $\gamma_R$ is a suitable vertical contour in the resolvent set; on the level of spectral integrals this inversion reduces to the elementary scalar inversion of Laplace transforms and is thus justified by dominated convergence.

**Trotter product and resolvent approximations.** The resolvent identity and functional calculus allow one to derive approximation formulas. For instance the exponential can be obtained as the strong limit of resolvent powers (Yosida approximants):

$$e^{tL} = \lim_{n \to \infty} \left(I - \frac{t}{n}L\right)^{-n}, \qquad (9.124)$$

which again follows by applying the scalar identity $(1 - \frac{t}{n}\lambda)^{-n} \to e^{t\lambda}$ for each $\lambda \in \sigma(L)$ together with dominated convergence under the spectral measure.

**Consequences and applications.** These spectral and resolvent relations are the analytic backbone of the Feynman-Kac correspondence: the pathwise representation of $e^{tL}$ as an expectation of multiplicative functionals is consistent with the spectral representation, and spectral properties (such as spectral gaps, eigenfunction expansions, trace-class criteria) may be translated into probabilistic statements (large-time decay, ground-state dominance, heat-kernel asymptotics). The functional calculus also enables perturbation analysis, spectral projections, and precise control of operator norms used throughout semigroup and Dirichlet-form theory.

## 9.7 Spectral Decomposition of Schrödinger Operators

Let $H$ denote the Schrödinger operator

$$H := -\frac{1}{2}\Delta + V(x) \qquad (9.125)$$

acting in the Hilbert space $L^2(\mathbb{R}^d)$, where the real-valued potential $V: \mathbb{R}^d \to \mathbb{R}$ satisfies $V(x) \ge V_0$ for some $V_0 \in \mathbb{R}$ and is chosen so that the quadratic form

$$\mathcal{E}^V(f, f) := \frac{1}{2} \int_{\mathbb{R}^d} |\nabla f(x)|^2 dx + \int_{\mathbb{R}^d} V(x) |f(x)|^2 dx, \quad f \in C_c^\infty(\mathbb{R}^d), \qquad (9.126)$$

is closable and lower semibounded. Passing to the closure $(\mathcal{E}^V, \mathcal{D}(\mathcal{E}^V))$ and applying the first representation theorem yields a unique self-adjoint operator $H$ (the Friedrichs extension of the formal differential expression) with

$$\mathcal{D}(\mathcal{E}^V) = \mathcal{D}((-H)^{1/2}), \quad \mathcal{E}^V(f, f) = \|(f)^{1/2}f\|_2^2, \quad f \in \mathcal{D}(\mathcal{E}^V). \qquad (9.127)$$

In particular $H$ is bounded from below and $\sigma(H) \subset [\lambda_{\inf}, \infty)$ for some $\lambda_{\inf} \in \mathbb{R}$.

By the spectral theorem for self-adjoint operators there exists a unique projection-valued measure $E^H(\cdot)$ on the Borel $\sigma$-algebra of $\mathbb{R}$ such that for any bounded Borel function $\varphi: \mathbb{R} \to \mathbb{C}$
the operator $\varphi(H)$ is defined by the spectral integral

$$ \varphi(H) = \int_{\mathbb{R}} \varphi(\lambda) dE^H(\lambda). \qquad (9.128) $$

Hence one obtains the spectral decomposition

$$ H = \int_{\mathbb{R}} \lambda dE^H(\lambda) \qquad (9.129) $$

and, for the heat (Schrödinger) semigroup,

$$ e^{-tH} = \int_{\mathbb{R}} e^{-t\lambda} dE^H(\lambda), \quad t \ge 0. \qquad (9.130) $$

The spectral measure $E^H$ decomposes the spectrum $\sigma(H)$ into mutually singular parts

$$ \sigma(H) = \sigma_{\text{pp}}(H) \cup \sigma_{\text{sc}}(H) \cup \sigma_{\text{ac}}(H), \qquad (9.131) $$

corresponding respectively to the pure point (eigenvalues), singular continuous, and absolutely continuous spectral components.

The semigroup representation yields the Laplace transform identity for matrix elements: for any $f, g \in L^2(\mathbb{R}^d)$ and $\alpha$ larger than the lower spectral bound,

$$ \langle (\alpha I + H)^{-1} f, g \rangle = \int_{0}^{\infty} e^{-\alpha t} \langle e^{-tH} f, g \rangle dt = \int_{\mathbb{R}} \frac{1}{\alpha + \lambda} d\langle E^H(\lambda) f, g \rangle, \qquad (9.132) $$

so that resolvent, semigroup and spectral measure are equivalent encodings of the operator $H$.

The Feynman-Kac formula furnishes a concrete probabilistic realization of the spectral measure via the semigroup kernel. Under hypotheses ensuring the Feynman-Kac representation (for example V Kato-class or form-bounded perturbation), the semigroup admits an integral kernel $K_t(x, y)$ satisfying

$$ (e^{-tH} f)(x) = \int_{\mathbb{R}^d} K_t(x, y) f(y) dy, \quad K_t(x, y) = p_t(x, y) \mathbb{E}_t^{x,y} \left[ e^{-\int_0^t V(X_s) ds} \right], \qquad (9.133) $$

where $p_t(x, y)$ is the Gaussian heat kernel and $\mathbb{E}_t^{x,y}$ denotes expectation with respect to the Brownian bridge from $x$ to $y$ over time $t$. The kernel $K_t(x, y)$ is related to the spectral measure by the inverse Laplace (or spectral) transform:

$$ K_t(x, y) = \int_{\mathbb{R}} e^{-t\lambda} d\mu_{x,y}^H(\lambda), \quad \mu_{x,y}^H(A) := \langle E^H(A)\delta_y, \delta_x \rangle, \qquad (9.134) $$

with $\delta_x$ denoting the Dirac distribution at $x$ (understood in the sense of distributional kernels when legitimate). Consequently, spectral properties (e.g. existence of a ground state, spectral gap, density of states) have immediate probabilistic manifestations in the long-time behaviour, trace asymptotics, and large deviations of the path measures.

When the semigroup $e^{-tH}$ is compact (for instance, due to confining growth of $V$ at infinity or working on a bounded domain with Dirichlet boundary), the spectrum is purely discrete
and one has a complete eigen-expansion. Concretely, if $\sigma(H) = \{\lambda_n\}_{n=1}^{\infty}$ with $\lambda_1 \le \lambda_2 \le \dots$ and $\lambda_n \to \infty$, and $\{\phi_n\}$ is an orthonormal basis of eigenfunctions, then

$$K_t(x, y) = \sum_{n=1}^{\infty} e^{-t\lambda_n} \phi_n(x) \overline{\phi_n(y)}, \qquad (9.135)$$

with the series converging uniformly for $t \ge t_0 > 0$ and in $L^2$-sense for each fixed $t > 0$. In particular,

$$(e^{-tH}f)(x) = \sum_{n=1}^{\infty} e^{-t\lambda_n} \langle f, \phi_n \rangle \phi_n(x). \qquad (9.136)$$

The ground state $\phi_1$ corresponding to the lowest eigenvalue $\lambda_1$ can be chosen strictly positive (Perron-Frobenius principle / positivity-improving property of $e^{-tH}$ when applicable), and spectral gap estimates $\lambda_2 - \lambda_1 > 0$ govern the exponential rate of convergence to equilibrium in $L^2$ and pointwise senses:

$$\lim_{t \to \infty} e^{t\lambda_1} K_t(x, y) = \phi_1(x) \overline{\phi_1(y)}. \qquad (9.137)$$

In the presence of absolutely continuous spectrum, one typically has no eigenfunction expansion but rather a spectral resolution in terms of generalized eigenfunctions or scattering states; the semigroup still admits the spectral integral representation

$$K_t(x, y) = \int_{\sigma(H)} e^{-t\lambda} \rho_{x,y}(\lambda) d\lambda, \qquad (9.138)$$

where $\rho_{x,y}$ denotes the density of the spectral measure with respect to Lebesgue measure on the absolutely continuous component. This representation underpins dispersive estimates and the short-time/large-energy asymptotics of the kernel via stationary phase and semiclassical analysis.

Trace-class and Schatten-class properties of $e^{-tH}$ are read off from the spectral side: $e^{-tH}$ is trace-class if and only if

$$\mathrm{Tr}(e^{-tH}) = \sum_{n} e^{-t\lambda_n} < \infty, \qquad (9.139)$$

which holds e.g. when $V$ is confining or when one restricts to compact domains. The trace yields the partition function in statistical mechanics and is connected to heat-kernel asymptotics and spectral invariants via the Minakshisundaram-Pleijel expansion or the Weyl law:

$$\mathrm{Tr}(e^{-tH}) \sim (4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{-tV(x)} dx + \dots, \quad t \downarrow 0, \qquad (9.140)$$

under suitable regularity/growth assumptions on $V$.

The functional calculus for $H$ permits one to translate analytic operations directly into probabilistic statements. For example, for $f \in L^2$ and measurable $\phi \ge 0$,

$$\langle \phi(H) f, f \rangle = \int_{\sigma(H)} \phi(\lambda) d\langle E^H(\lambda) f, f \rangle = \int_{0}^{\infty} \phi(\lambda) d\mu_{f}^{H}(\lambda), \qquad (9.141)$$

and when $\phi(\lambda) = e^{-t\lambda}$ the right-hand side equals $\langle e^{-tH}f, f \rangle = \mathbb{E}[\dots]$ via Feynman-Kac. This equivalence is the cornerstone of the analytic-probabilistic duality: spectral information (eigen-
values, spectral measures, resolvent poles) corresponds to pathwise phenomena (long-time decay, ground-state dominance, resonance behaviour).

Finally, perturbation theory (Kato–Rellich, Birman–Schwinger) and semiclassical analysis connect fine spectral features of $H$ to the small-noise / large-parameter asymptotics of the path integrals. For instance, the Birman–Schwinger principle relates eigenvalues below the essential spectrum to compactness properties of a suitable integral operator built from the free resolvent and the potential $V$, while semiclassical (WKB) methods approximate high-energy spectral densities by phase-space integrals, which also admit probabilistic interpretations via large-deviation principles for the Brownian motion path measures.

In summary, the spectral decomposition of $H$ and the Feynman–Kac representation are two equivalent and complementary lenses: the former is the algebraic/spectral description furnished by functional calculus, and the latter is the analytic/probabilistic description furnished by expectations over path space. Each perspective supplies tools and estimates that translate directly into the other, yielding a rich and exact correspondence between spectral theory, PDE analysis, potential theory, and stochastic processes.

## 9.8 Conclusion

In this chapter, we examined the functional analytic structure underlying Feynman–Kac semigroups and their generators. The discussion unified semigroup theory, spectral theory, and Sobolev space analysis to characterize the mapping and regularity properties of the associated operators.

We began by investigating the domain of the generator $H = -\frac{1}{2}\Delta_g + V$ as a self-adjoint operator on $L^2(M, dvol_g)$, ensuring that $H$ admits a well-defined spectral decomposition. Using this, we connected the Feynman–Kac semigroup $e^{-tH}$ to the spectral resolution of $H$ via

$$e^{-tH} = \int_{0}^{\infty} e^{-t\lambda} dE_{\lambda}, \qquad (9.142)$$

where $E_{\lambda}$ is the spectral measure. This relation provided a precise quantitative framework for understanding diffusion smoothing, spectral gaps, and the asymptotic decay of heat kernels. We then analyzed compactness, ultracontractivity, and trace-class properties of $e^{-tH}$, relating them to geometric quantities such as Ricci curvature and volume growth. The equivalence between functional inequalities (Poincaré, log-Sobolev) and analytic semigroup properties was established, showing that curvature bounds yield precise $L^p$-regularization and entropy dissipation estimates.

In conclusion, this chapter provided the deepest analytic insight into how geometry, probability, and spectral theory intertwine in the study of diffusion semigroups. Through the functional analytic lens, the Feynman–Kac representation emerges as a unifying theorem connecting path expectations, operator theory, and geometric analysis on manifolds.

**References:** Reed and Simon (1980) [21], Davies (1989) [13], DaPrato and Zabczyk (1992) [10].
# Part IV

## Generalizations and Modern Extensions
# Extensions to Lévy and Jump Processes

> "Every discontinuity hides a structure; every jump conceals a law."
> 
> — Paul Lévy

## 10.1 Motivation and General Framework

The analysis of diffusion semigroups and their generators naturally extends to stochastic processes with jumps. While Brownian motion yields continuous trajectories governed by local operators such as the Laplacian, jump processes correspond to nonlocal generators — operators involving integro-differential terms. This framework unifies diffusion and jump phenomena under the general theory of Lévy processes, which serve as the building blocks of all stochastic processes with stationary independent increments.

A Lévy process $(X_t)_{t \ge 0}$ on $\mathbb{R}^d$ is characterized by three components: a drift vector $b \in \mathbb{R}^d$, a covariance matrix $Q \in \mathbb{R}^{d \times d}$, and a Lévy measure $\nu$ on $\mathbb{R}^d \setminus \{0\}$ satisfying

$$ \int_{\mathbb{R}^d \setminus \{0\}} \min(1, |x|^2) \nu(dx) < \infty. \qquad (10.1) $$

These data determine both the infinitesimal generator of the process and the structure of its sample paths.

## 10.2 Lévy–Khintchine Representation

The Lévy–Khintchine representation theorem provides the most general analytic characterization of infinitely divisible probability distributions and, equivalently, of Lévy processes. A Lévy process $(X_t)_{t \ge 0}$ on $\mathbb{R}^d$ is a stochastic process satisfying:

$$ X_0 = 0 \text{ a.s., } X_t - X_s \text{ is independent of } \mathcal{F}_s, X_t - X_s \sim X_{t-s}, \text{ and } X_t \text{ is stochastically continuous.} \qquad (10.2) $$

That is, $X_t$ has stationary and independent increments with càdlàg sample paths.

For such a process, each random variable $X_t$ is infinitely divisible, meaning that for any $n \in \mathbb{N}$,
there exist i.i.d. random variables $Y_1, \dots, Y_n$ such that

$$X_t \stackrel{d}{=} Y_1 + \dots + Y_n. \qquad (10.3)$$

Hence, the characteristic function $\phi_t(\xi) = \mathbb{E}[e^{i\xi \cdot X_t}]$ satisfies

$$\phi_t(\xi) = e^{-t\psi(\xi)}, \quad \xi \in \mathbb{R}^d, \qquad (10.4)$$

where $\psi: \mathbb{R}^d \to \mathbb{C}$ is called the characteristic exponent or Lévy symbol. The mapping $t \mapsto \phi_t(\xi)$ satisfies the semigroup property

$$\phi_{s+t}(\xi) = \phi_s(\xi)\phi_t(\xi), \qquad (10.5)$$

which implies that $\psi(\xi)$ uniquely determines the process.

The Lévy-Khintchine formula asserts that any function $\psi(\xi)$ of the form

$$\psi(\xi) = ib \cdot \xi + \frac{1}{2}\xi^T Q \xi + \int_{\mathbb{R}^d \setminus \{0\}} \left(1 - e^{i\xi \cdot x} + i\xi \cdot x \mathbf{1}_{\{|x|<1\}}\right) \nu(dx) \qquad (10.6)$$

corresponds to the characteristic exponent of a Lévy process, and conversely, every Lévy process admits such a representation. Here:

* $b \in \mathbb{R}^d$ is the drift vector;

* $Q \in \mathbb{R}^{d \times d}$ is a symmetric nonnegative-definite matrix representing the Gaussian covariance;

* $\nu$ is a Lévy measure on $\mathbb{R}^d \setminus \{0\}$ satisfying

$$\int_{\mathbb{R}^d \setminus \{0\}} \min(1, |x|^2) \nu(dx) < \infty. \qquad (10.7)$$

The three terms in $\psi(\xi)$ have distinct probabilistic interpretations:

1. $ib \cdot \xi$ corresponds to the deterministic drift component;

2. $\frac{1}{2}\xi^T Q \xi$ corresponds to the Gaussian diffusion component;

3. the integral term represents the jump component, describing random discontinuities with intensity governed by $\nu$.

The Lévy measure $\nu$ encodes the frequency and magnitude of jumps, with small jumps regularized by the term $i\xi \cdot x \mathbf{1}_{\{|x|<1\}}$ to ensure convergence of the integral. If $\nu = 0$, the process reduces to a Gaussian process with drift and covariance $Q$.

The infinitesimal generator $L$ of the Lévy process, acting on test functions $f \in C_c^\infty(\mathbb{R}^d)$, is then given by the Lévy-Itô operator:

$$(Lf)(x) = b \cdot \nabla f(x) + \frac{1}{2} \mathrm{Tr}(Q \nabla^2 f(x)) + \int_{\mathbb{R}^d \setminus \{0\}} (f(x+y) - f(x) - y \cdot \nabla f(x) \mathbf{1}_{\{|y|<1\}}) \nu(dy). \quad (10.8)$$
This operator is the infinitesimal generator of the strongly continuous contraction semigroup $(P_t)_{t \ge 0}$ defined by

$$ (P_t f)(x) = \mathbb{E}^x[f(X_t)] = \int_{\mathbb{R}^d} f(x+y) p_t(dy), \qquad (10.9) $$

where $p_t$ is the transition probability measure of $X_t$.

Finally, the spectral symbol of the generator in Fourier space satisfies

$$ \hat{L}f(\xi) = -\psi(\xi) \hat{f}(\xi), \qquad (10.10) $$

showing that $L$ acts as a pseudo-differential operator with symbol $-\psi(\xi)$. Hence, the Lévy-Khintchine formula not only characterizes the law of Lévy processes but also provides the analytic bridge between probability theory and harmonic analysis via the spectral representation of nonlocal generators.

## 10.3 Nonlocal Dirichlet Forms

In the context of symmetric Lévy processes, the infinitesimal generator is inherently nonlocal, reflecting the possibility of instantaneous jumps from one spatial location to another. The associated bilinear Dirichlet form captures this nonlocality through its dependence on spatial increments rather than derivatives.

Let $X_t$ be a symmetric Lévy process on $\mathbb{R}^d$ with Lévy triplet $(0, 0, \nu)$, where $\nu$ is a symmetric Lévy measure satisfying

$$ \nu(A) = \nu(-A), \quad \forall A \in \mathcal{B}(\mathbb{R}^d \setminus \{0\}), \qquad (10.11) $$

and

$$ \int_{\mathbb{R}^d \setminus \{0\}} \min(1, |y|^2) \nu(dy) < \infty. \qquad (10.12) $$

The symmetry condition implies that the process $X_t$ has no drift or Gaussian component and that its transition probabilities are symmetric, i.e.,

$$ p_t(x, y) = p_t(y, x), \quad \forall x, y \in \mathbb{R}^d. \qquad (10.13) $$

The corresponding Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ acting on $L^2(\mathbb{R}^d, dx)$ is defined by

$$ \mathcal{E}(f, g) = \frac{1}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} (f(x+y) - f(x))(g(x+y) - g(x)) \nu(dy) dx, \qquad (10.14) $$

with the form domain

$$ \mathcal{D}(\mathcal{E}) = \left\{ f \in L^2(\mathbb{R}^d) : \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |f(x+y) - f(x)|^2 \nu(dy) dx < \infty \right\}. \qquad (10.15) $$

This form arises from the $L^2$-limit of the energy associated with the process, and it satisfies the following fundamental analytic properties:
**Symmetry:** For all $f, g \in \mathcal{D}(\mathcal{E})$,

$$ \mathcal{E}(f, g) = \mathcal{E}(g, f), \qquad (10.16) $$

which follows directly from the symmetry of the Lévy measure $\nu(A) = \nu(-A)$ and the change of variables $y \mapsto -y$ in the integral.

**Positivity:** For all $f \in \mathcal{D}(\mathcal{E})$,

$$ \mathcal{E}(f, f) = \frac{1}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} |f(x + y) - f(x)|^2 \nu(dy) dx \geq 0, \qquad (10.17) $$

so that the form is nonnegative definite.

**Closedness:** The form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is closed, meaning that if $(f_n)_{n \in \mathbb{N}} \subset \mathcal{D}(\mathcal{E})$ satisfies

$$ f_n \to f \text{ in } L^2(\mathbb{R}^d) \quad \text{and} \quad \mathcal{E}(f_n - f_m, f_n - f_m) \to 0 \text{ as } n, m \to \infty, \qquad (10.18) $$

then $f \in \mathcal{D}(\mathcal{E})$ and $\mathcal{E}(f_n - f, f_n - f) \to 0$. This follows from Fatou's lemma and the completeness of $L^2(\mathbb{R}^d)$.

**Markov Property:** For any $f \in \mathcal{D}(\mathcal{E})$ and the normal contraction $\tilde{f} = (0 \lor f) \land 1$, one has $\tilde{f} \in \mathcal{D}(\mathcal{E})$ and

$$ \mathcal{E}(\tilde{f}, \tilde{f}) \leq \mathcal{E}(f, f). \qquad (10.19) $$

Thus, $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is a Dirichlet form in the sense of Fukushima, Oshima, and Takeda.

**Associated Nonlocal Generator:** The self-adjoint operator $L$ associated with $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ via the first representation theorem satisfies

$$ \mathcal{E}(f, g) = \langle -Lf, g \rangle_{L^2(\mathbb{R}^d)}, \quad f \in \mathcal{D}(L), g \in \mathcal{D}(\mathcal{E}), \qquad (10.20) $$

and for $f \in C_c^\infty(\mathbb{R}^d)$,

$$ (Lf)(x) = \int_{\mathbb{R}^d} (f(x + y) - f(x)) \nu(dy), \qquad (10.21) $$

or, after compensation for small jumps (ensuring convergence),

$$ (Lf)(x) = \int_{\mathbb{R}^d \setminus \{0\}} (f(x + y) - f(x) - y \cdot \nabla f(x)) \mathbf{1}_{\{|y|<1\}} \nu(dy). \qquad (10.22) $$

**Fourier Representation:** By Parseval's identity, one obtains the equivalent spectral characterization:

$$ \mathcal{E}(f, f) = \int_{\mathbb{R}^d} \psi(\xi) |\hat{f}(\xi)|^2 d\xi, \qquad (10.23) $$

where $\psi(\xi)$ is the characteristic exponent associated with $\nu$,

$$ \psi(\xi) = \int_{\mathbb{R}^d \setminus \{0\}} (1 - \cos(\xi \cdot y)) \nu(dy). \qquad (10.24) $$

Hence, the Dirichlet form $\mathcal{E}$ fully captures the nonlocal jump structure of the process in both spatial and Fourier domains. By the general theory of Dirichlet forms, $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ generates
a symmetric Hunt process with càdlàg trajectories and jump kernel $\nu$, thereby providing a complete analytic characterization of symmetric pure-jump Lévy processes.

## 10.4 Fractional Laplacian as a Paradigm

A central and paradigmatic example of a nonlocal generator arising from a symmetric Lévy process is the fractional Laplacian $(-\Delta)^{\alpha/2}$, where the order of differentiation $\alpha \in (0, 2)$ determines the jump intensity and tail behavior of the corresponding Lévy process. This operator generalizes the classical Laplacian to the nonlocal setting and plays a fundamental role in probability theory, harmonic analysis, and the theory of partial differential equations.

**Definition via Fourier Symbol:** For $f \in S(\mathbb{R}^d)$, the Schwartz space of rapidly decaying smooth functions, the fractional Laplacian is defined in the Fourier domain by

$$ \mathcal{F}[(-\Delta)^{\alpha/2} f](\xi) = |\xi|^{\alpha} \hat{f}(\xi), \qquad (10.25) $$

where

$$ \hat{f}(\xi) = \int_{\mathbb{R}^d} e^{-i\xi \cdot x} f(x) dx \qquad (10.26) $$

is the Fourier transform of $f$. This definition exhibits the operator as a pseudodifferential operator with symbol $|\xi|^{\alpha}$, showing that $(-\Delta)^{\alpha/2}$ is positive, self-adjoint, and homogeneous of order $\alpha$.

**Singular Integral Representation:** The same operator admits a real-space representation as a singular integral:

$$ (-\Delta)^{\alpha/2} f(x) = C_{d,\alpha} \text{P.V.} \int_{\mathbb{R}^d} \frac{f(x) - f(y)}{|x-y|^{d+\alpha}} dy, \qquad (10.27) $$

where P.V. denotes the Cauchy principal value and the normalization constant $C_{d,\alpha} > 0$ is explicitly given by

$$ C_{d,\alpha} = \frac{2^{\alpha-1}\alpha \Gamma(\frac{d+\alpha}{2})}{\pi^{d/2}\Gamma(1-\frac{\alpha}{2})}. \qquad (10.28) $$

This normalization ensures the identity

$$ \mathcal{F}[(-\Delta)^{\alpha/2} f](\xi) = |\xi|^{\alpha} \hat{f}(\xi) \qquad (10.29) $$

for all $f \in S(\mathbb{R}^d)$. The integral form reveals the nonlocality of $(-\Delta)^{\alpha/2}$: the value of the operator at a point $x$ depends on all values of $f$ throughout $\mathbb{R}^d$, with the kernel $|x-y|^{-d-\alpha}$ governing the strength of long-range interactions.

**Dirichlet Form Representation:** The corresponding bilinear Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ associated with the fractional Laplacian is defined by

$$ \mathcal{E}(f,g) = \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(f(x)-f(y))(g(x)-g(y))}{|x-y|^{d+\alpha}} dx dy, \qquad (10.30) $$
with form domain

$$ \mathcal{D}(\mathcal{E}) = \left\{ f \in L^2(\mathbb{R}^d) : \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy < \infty \right\}. \qquad (10.31) $$

The form $\mathcal{E}$ is symmetric, nonnegative, and closed, and hence by the first representation theorem there exists a unique self-adjoint operator $L = -(-\Delta)^{\alpha/2}$ such that

$$ \mathcal{E}(f, g) = \langle (-L)^{1/2} f, (-L)^{1/2} g \rangle_{L^2(\mathbb{R}^d)}. \qquad (10.32) $$

**Energy Identity and Spectral Representation:** In the Fourier domain, the energy functional can be equivalently expressed as

$$ \mathcal{E}(f, f) = \int_{\mathbb{R}^d} |\xi|^\alpha |\hat{f}(\xi)|^2 d\xi. \qquad (10.33) $$

This equality follows from Plancherel's theorem and provides the spectral characterization of the fractional Laplacian as the multiplication operator with symbol $|\xi|^\alpha$. It also implies that $(-\Delta)^{\alpha/2}$ is the infinitesimal generator of an $\alpha$-stable Lévy process, whose characteristic exponent is precisely $\psi(\xi) = |\xi|^\alpha$.

**Probabilistic Interpretation:** Let $X_t$ be a rotationally symmetric $\alpha$-stable Lévy process on $\mathbb{R}^d$ with characteristic function

$$ \mathbb{E}[e^{i\xi \cdot X_t}] = e^{-t|\xi|^\alpha}, \quad \xi \in \mathbb{R}^d. \qquad (10.34) $$

Then the infinitesimal generator of the associated semigroup $(T_t)_{t \ge 0}$, given by

$$ (T_t f)(x) = \mathbb{E}_x[f(X_t)] = \int_{\mathbb{R}^d} p_t(x - y)f(y) dy, \qquad (10.35) $$

is exactly $L = -(-\Delta)^{\alpha/2}$. Thus, the fractional Laplacian serves as the analytic generator of the isotropic $\alpha$-stable process, linking nonlocal analysis and jump-type stochastic dynamics.

**Variational Characterization:** The space $\mathcal{D}(\mathcal{E})$, endowed with the inner product

$$ \langle f, g \rangle_{\mathcal{E}} = \langle f, g \rangle_{L^2(\mathbb{R}^d)} + \mathcal{E}(f, g), \qquad (10.36) $$

is a Hilbert space coinciding with the fractional Sobolev space $H^{\alpha/2}(\mathbb{R}^d)$. The energy form $\mathcal{E}$ thus represents the quadratic form associated with $(-\Delta)^{\alpha/2}$, providing both an analytic and probabilistic characterization of the fractional Laplacian.

Hence, $(-\Delta)^{\alpha/2}$ stands as the canonical prototype of a nonlocal, self-adjoint, and Markovian operator, embodying the structure of symmetric $\alpha$-stable Lévy generators and their associated Dirichlet forms.
## 10.5 Feynman-Kac Representation with Jumps

In the nonlocal and jump-diffusion setting, the classical Feynman-Kac formula extends naturally to operators whose generators include both differential and integro-differential components. Let $X_t$ be a symmetric Lévy process on $\mathbb{R}^d$ with generator $L$ acting on $C_c^\infty(\mathbb{R}^d)$ by

$$ (Lf)(x) = b \cdot \nabla f(x) + \frac{1}{2} \mathrm{Tr}(Q \nabla^2 f(x)) + \int_{\mathbb{R}^d \setminus \{0\}} (f(x+y) - f(x) - y \cdot \nabla f(x) \mathbf{1}_{\{|y|<1\}}) \nu(dy), \quad (10.37) $$

where $b \in \mathbb{R}^d$ is the drift vector, $Q$ is a positive semidefinite covariance matrix, and $\nu$ is the Lévy measure satisfying

$$ \int_{\mathbb{R}^d \setminus \{0\}} \min(1, |y|^2) \nu(dy) < \infty. \quad (10.38) $$

The process $X_t$ has càdlàg (right-continuous with left limits) paths in the Skorokhod space $D([0, \infty), \mathbb{R}^d)$, and it can be decomposed as

$$ X_t = bt + W_t + \int_0^t \int_{|y|<1} y (\tilde{N}(ds, dy)) + \int_0^t \int_{|y|\ge 1} y N(ds, dy), \quad (10.39) $$

where $W_t$ is a Brownian motion, $N(ds, dy)$ is the Poisson random measure of jumps, and $\tilde{N}(ds, dy) = N(ds, dy) - \nu(dy)ds$ is the compensated Poisson measure.

**Schrödinger-type Nonlocal Operators:** Let $V : \mathbb{R}^d \to \mathbb{R}$ be a measurable potential function, bounded from below or belonging to the Kato class associated with $X_t$. We consider the nonlocal Schrödinger operator

$$ H = -L + V, \quad (10.40) $$

initially defined on $C_c^\infty(\mathbb{R}^d)$, and then extended to a self-adjoint operator on $L^2(\mathbb{R}^d)$ via the Friedrichs extension of the associated quadratic form

$$ \mathcal{E}^V(f, f) = \mathcal{E}(f, f) + \int_{\mathbb{R}^d} V(x) |f(x)|^2 dx, \quad (10.41) $$

where $\mathcal{E}$ is the symmetric Dirichlet form corresponding to the Lévy process.

**Generalized Feynman-Kac Formula:** The semigroup $(e^{-tH})_{t \ge 0}$ generated by $H$ admits a probabilistic representation in terms of expectations over the Lévy process:

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \quad f \in L^2(\mathbb{R}^d). \quad (10.42) $$

This equality defines the generalized Feynman-Kac formula with jumps. The expectation is taken with respect to the law of the process $X_t$ starting at $x$, and the exponential weight incorporates the pathwise accumulated potential energy $\int_0^t V(X_s) ds$.

**Analytic Foundation:** From the analytic perspective, the mapping

$$ P_t^V f(x) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right] \quad (10.43) $$
defines a strongly continuous contraction semigroup on $L^2(\mathbb{R}^d)$ under suitable assumptions on $V$, such as the Kato smallness condition

$$ \limsup_{t \downarrow 0} \sup_{x \in \mathbb{R}^d} \mathbb{E}^x \left[ \int_0^t |V(X_s)| ds \right] = 0. \qquad (10.44) $$

By the general semigroup theory, its generator coincides with the self-adjoint realization of $H = -L + V$. Hence,

$$ P_t^V = e^{-tH}, \quad \forall t \ge 0. \qquad (10.45) $$

**Path Integral over Skorokhod Space:** The probabilistic representation can be rigorously viewed as an expectation over the Skorokhod path space $D([0, t], \mathbb{R}^d)$ endowed with the canonical measure induced by the Lévy process:

$$ (e^{-tH} f)(x) = \int_{D([0,t],\mathbb{R}^d)} e^{-\int_0^t V(\omega(s)) ds} f(\omega(t)) \mathbb{P}^x(d\omega). \qquad (10.46) $$

This integral defines a nonlocal analogue of the classical Feynman path integral, where the trajectories $\omega$ include both continuous Brownian paths and jump discontinuities governed by $\nu$.

**Analytic-Probabilistic Duality:** The above representation preserves the fundamental duality between analytic and probabilistic descriptions:

* The analytic side is represented by the semigroup $e^{-tH}$ acting on $L^2(\mathbb{R}^d)$, with infinitesimal generator $H = -L + V$.

* The probabilistic side is given by the expectation over Lévy trajectories, with the exponential weight incorporating the potential energy functional.

Even in the presence of jumps, this duality remains exact, thus extending the classical Feynman-Kac correspondence to the realm of nonlocal operators and discontinuous stochastic processes.

## 10.6 Generators with Drift and Jump-Diffusion Operators

In the most general setting of Markov processes with both continuous and discontinuous components, one encounters *jump-diffusion generators* that combine drift, diffusion, and jump contributions in a single integro-differential operator. These generators characterize the infinitesimal behavior of a large class of Itô semimartingales, encompassing both local (diffusion) and nonlocal (jump) phenomena.

**General Form of the Generator:** Let $L$ be an operator acting on sufficiently smooth functions $f \in C_c^2(\mathbb{R}^d)$ by

$$ (Lf)(x) = b(x) \cdot \nabla f(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x) + \int_{\mathbb{R}^d \setminus \{0\}} \left( f(x+y) - f(x) - y \cdot \nabla f(x) \mathbf{1}_{\{|y|<1\}} \right) \nu(x, dy), \qquad (10.47) $$

where:

* $b: \mathbb{R}^d \to \mathbb{R}^d$ is a measurable *drift field* describing the deterministic first-order motion;
* $a(x) = (a_{ij}(x))_{1\le i,j\le d}$ is a measurable, symmetric, positive semidefinite matrix field representing the local diffusion coefficients;

* $\nu(x, dy)$ is a position-dependent Lévy kernel (or jump measure) satisfying the integrability condition

$$ \int_{\mathbb{R}^d \setminus \{0\}} (1 \wedge |y|^2) \nu(x, dy) < \infty, \quad \forall x \in \mathbb{R}^d. \qquad (10.48) $$

The truncation term $y \cdot \nabla f(x) \mathbf{1}_{\{|y|<1\}}$ guarantees the convergence of the integral near $y = 0$ by compensating the linear part of small jumps, thus ensuring that $Lf(x)$ is well-defined for twice continuously differentiable $f$.

**Probabilistic Representation via Stochastic Differential Equations:** The operator $L$ in (10.47) arises as the infinitesimal generator of a jump-diffusion process $(X_t)_{t\ge0}$ satisfying the stochastic differential equation

$$ dX_t = b(X_{t-}) dt + \sigma(X_{t-}) dW_t + \int_{|y|<1} y \tilde{N}(dt, dy) + \int_{|y|\ge1} y N(dt, dy), \qquad (10.49) $$

where:

* $W_t$ is a standard $d$-dimensional Brownian motion;

* $\sigma(x)$ is a measurable diffusion coefficient matrix satisfying $a(x) = \sigma(x)\sigma(x)^T$;

* $N(dt, dy)$ is a Poisson random measure on $(0, \infty) \times (\mathbb{R}^d \setminus \{0\})$ with compensator $\nu(X_{t-}, dy) dt$;

* $\tilde{N}(dt, dy) = N(dt, dy) - \nu(X_{t-}, dy) dt$ denotes the compensated Poisson measure, ensuring martingale properties for small jumps.

The process $X_t$ is a càdlàg semimartingale taking values in $\mathbb{R}^d$, and its infinitesimal generator is precisely $L$ defined in (10.47). Formally, for all $f \in C_c^2(\mathbb{R}^d)$,

$$ f(X_t) - f(X_0) - \int_0^t (Lf)(X_s) ds \qquad (10.50) $$

is a martingale with respect to the natural filtration of $X_t$.

**Analytic Properties:** The operator $L$ is generally non-symmetric because of the drift term $b(x) \cdot \nabla f(x)$ and the possible asymmetry of $\nu(x, dy)$. However, it retains several essential analytic properties:

* $L$ is local in the diffusion and drift parts but nonlocal in the jump component.

* When $b(x) \equiv 0$, $a(x) \equiv 0$, and $\nu(x, dy) = \nu(dy)$ is symmetric, $L$ reduces to the generator of a pure Lévy process, often associated with a Dirichlet form.

* When $\nu(x, dy) \equiv 0$, $L$ reduces to a second-order differential operator with drift, corresponding to an Itô diffusion.

The combination of these features makes $L$ a unifying operator encompassing both continuous and jump-driven dynamics.
**Domain and Regularity Considerations:** To ensure that $Lf(x)$ is well-defined and belongs to $L_{\text{loc}}^2(\mathbb{R}^d)$, one typically requires that

$$b(\cdot) \in L_{\text{loc}}^p(\mathbb{R}^d; \mathbb{R}^d), \quad a_{ij}(\cdot) \in L_{\text{loc}}^{\infty}(\mathbb{R}^d), \quad \text{and} \quad \nu(x, dy) \text{ measurable in } x. \qquad (10.51)$$

If these conditions hold, the associated operator $L$ generates a (possibly non-symmetric) strongly continuous semigroup $(P_t)_{t \ge 0}$ on $L^2(\mathbb{R}^d)$, which acts via

$$(P_t f)(x) = \mathbb{E}^x[f(X_t)]. \qquad (10.52)$$

**Interpretation and Applications:** The generator (10.47) serves as the mathematical foundation for a wide variety of models:

* In mathematical finance, it appears in jump-diffusion models of asset prices (e.g., Merton's model, Kou's double-exponential jump diffusion).

* In physics, it describes transport phenomena with both diffusive and ballistic (jump) components.

* In biology and ecology, it models spatial population dynamics with long-range dispersal.

Hence, the operator $L$ provides a unifying analytical framework for stochastic dynamics that exhibit both continuous diffusion and discontinuous jumps, bridging the theory of partial differential equations and the probabilistic theory of semimartingales.

## 10.7 Analytic and Probabilistic Duality

The profound connection between nonlocal analytic operators and stochastic processes is established through the framework of *Dirichlet forms* and their generalizations. When symmetry fails or nonlocal interactions dominate, one must extend the classical Dirichlet form theory to the broader setting of *semi-Dirichlet forms* and *sectorial operators*, which retain enough analytic structure to ensure the existence of an associated Markov process.

**Semi-Dirichlet and Sectorial Forms:** Let $E$ be a locally compact separable metric space with a positive Radon measure $m$ having full support. A bilinear form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(E; m)$ is called a *semi-Dirichlet form* if it satisfies the following conditions:

1. **Densely defined:** The domain $\mathcal{D}(\mathcal{E})$ is dense in $L^2(E; m)$.

2. **Closedness:** For any sequence $(f_n) \subset \mathcal{D}(\mathcal{E})$ with $f_n \to f$ in $L^2(E; m)$ and $\mathcal{E}(f_n - f_m, f_n - f_m) \to 0$, we have $f \in \mathcal{D}(\mathcal{E})$ and $\mathcal{E}(f_n - f, f_n - f) \to 0$.

3. **Sector condition:** There exists a constant $K \ge 0$ such that for all $f, g \in \mathcal{D}(\mathcal{E})$,

$$|\mathcal{E}(f, g) - \mathcal{E}(g, f)| \le K [\mathcal{E}(f, f)^{1/2} \mathcal{E}(g, g)^{1/2}]. \qquad (10.53)$$

This ensures that the form is *sectorial*, i.e., its numerical range lies in a sector of the complex plane.
4. **Markov property:** For every $f \in \mathcal{D}(\mathcal{E})$, the truncated function $f^{\#} := (0 \lor f) \land 1$ also belongs to $\mathcal{D}(\mathcal{E})$, and

$$ \mathcal{E}(f^{\#}, f^{\#}) \leq \mathcal{E}(f, f). \qquad (10.54) $$

The pair $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ is then associated with a (not necessarily symmetric) strongly continuous semigroup $(T_t)_{t \ge 0}$ on $L^2(E; m)$ satisfying the sectorial bound

$$ \|T_t\|_{L^2 \to L^2} \le e^{Kt}, \quad t \ge 0. \qquad (10.55) $$

**Beurling–Deny Representation:** The central analytic structure underlying nonlocal and jump processes is provided by the *Beurling–Deny decomposition*, which expresses any regular semi-Dirichlet form as a sum of three mutually orthogonal components:

$$ \mathcal{E}(f,g) = \mathcal{E}^{(c)}(f,g) + \int_{E \times E \setminus \text{diag}} (f(x)-f(y))(g(x)-g(y)) J(dx, dy) + \int_E f(x)g(x) k(dx), \quad (10.56) $$

for all $f, g \in \mathcal{D}(\mathcal{E})$, where:

* $\mathcal{E}^{(c)}$ is the local (continuous) part corresponding to diffusion; it is strongly local in the sense that

$$ \mathcal{E}^{(c)}(f,g) = 0 \quad \text{whenever } f \text{ is constant on a neighborhood of } \text{supp}(g); \qquad (10.57) $$

* $J(dx, dy)$ is a symmetric, positive Radon measure on $E \times E \setminus \text{diag}$, referred to as the *jump measure*, encoding the intensity and distribution of discontinuous jumps;

* $k(dx)$ is a positive Radon measure on $E$ known as the *killing measure*, representing the rate at which the process is terminated (or killed).

The form (10.56) thus decomposes any semi-Dirichlet form into diffusion, jump, and killing contributions—an exact analytic counterpart of the drift-diffusion-jump structure of general stochastic generators.

**Correspondence with Markov Processes:** For a regular semi-Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(E; m)$, there exists a unique (up to $m$-equivalence) *Hunt process* $(X_t)_{t \ge 0}$ such that:

$$ T_t f(x) = \mathbb{E}^x[f(X_t)], \quad \forall f \in L^2(E; m), \qquad (10.58) $$

where $(T_t)_{t \ge 0}$ is the semigroup associated with $\mathcal{E}$. This establishes the *analytic-probabilistic duality*: every semi-Dirichlet form corresponds to a (possibly non-symmetric) Markov process, and conversely, every such process admits an energy form of the Beurling–Deny type.

**Relation to the Lévy–Khintchine Structure:** The decomposition (10.56) generalizes the Lévy–Khintchine representation of infinitely divisible processes. Indeed:

* the local part $\mathcal{E}^{(c)}$ corresponds to the Gaussian (Brownian) component;

* the jump measure $J(dx, dy)$ corresponds to the Lévy measure $\nu(dy)$;

* the killing term $k(dx)$ corresponds to an exponential killing rate or absorption potential.
Hence, the Beurling–Deny representation unifies both local (differential) and nonlocal (integral) aspects of stochastic dynamics within a single variational framework.

**Fundamental Duality Principle:** The analytic form and its associated stochastic process are two realizations of the same underlying object:

$$ \text{Energy form } \mathcal{E}(f,g) \leftrightarrow \text{Infinitesimal generator } L, \leftrightarrow \text{Hunt process } (X_t)_{t \ge 0}. \quad (10.59) $$

This triad expresses the deep equivalence between the variational, analytic, and probabilistic formulations of nonlocal stochastic dynamics, extending classical diffusion theory to the general setting of Lévy-type and jump processes.

## 10.8 Spectral and Regularity Properties

The spectral and regularity theory of nonlocal operators such as the fractional Laplacian and, more generally, Lévy-type generators, forms one of the central pillars of modern nonlocal analysis. Although these operators share deep analogies with their local elliptic counterparts, their spectral structure and the nature of regularity of their eigenfunctions differ in subtle but fundamental ways, owing to the nonlocal interactions encoded in their jump kernels.

**Spectral Structure and Discreteness:** Let $L = (-\Delta)^{\alpha/2} + V(x)$ on $L^2(\mathbb{R}^d)$, with $0 < \alpha < 2$, where $V: \mathbb{R}^d \to [0, \infty)$ is a measurable confining potential satisfying

$$ V(x) \to +\infty \quad \text{as} \quad |x| \to \infty. \quad (10.60) $$

Under these conditions, $L$ defines a self-adjoint, nonnegative operator on $L^2(\mathbb{R}^d)$ with form domain

$$ \mathcal{D}(\mathcal{E}_V) = \left\{ f \in L^2(\mathbb{R}^d) : \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy + \int_{\mathbb{R}^d} V(x) |f(x)|^2 dx < \infty \right\}. \quad (10.61) $$

The associated quadratic form

$$ \mathcal{E}_V(f, f) = C_{d,\alpha} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy + \int_{\mathbb{R}^d} V(x) |f(x)|^2 dx \quad (10.62) $$

is closed, symmetric, and coercive. By standard compactness arguments (Rellich-type embedding theorems for fractional Sobolev spaces), one deduces that the resolvent $(L + \lambda I)^{-1}$ is compact on $L^2(\mathbb{R}^d)$ for all $\lambda > 0$. Consequently, $L$ admits a purely discrete spectrum:

$$ 0 \le \lambda_1 < \lambda_2 \le \lambda_3 \le \dots, \quad \lambda_n \to +\infty, \quad (10.63) $$

and the corresponding eigenfunctions $\{\phi_n\}_{n \ge 1}$ form a complete orthonormal basis of $L^2(\mathbb{R}^d)$.

**Regularity of Eigenfunctions:** Unlike classical elliptic operators, eigenfunctions of nonlocal operators are not smooth in general, due to the singular and integral nature of the generator. However, they enjoy a form of fractional regularity.
Precisely, if $L = (-\Delta)^{\alpha/2} + V(x)$ with $V \in L_{\text{loc}}^{\infty}(\mathbb{R}^d)$, and $L\phi = \lambda\phi$, then by fractional elliptic regularity theory, one has

$$ \phi \in C_{\text{loc}}^{\alpha+\beta}(\mathbb{R}^d) \quad \text{for some } \beta \in (0, 1-\alpha), \qquad (10.64) $$

and moreover, if $V$ is Hölder continuous, one obtains global Hölder continuity

$$ \phi \in C^{\alpha+\gamma}(\mathbb{R}^d) \quad \text{for some } \gamma > 0. \qquad (10.65) $$

This fractional regularity reflects the nonlocal smoothing properties of the operator: jumps connect distant regions of the space, propagating regularity in a weaker, integral sense rather than through differential smoothing.

**Heat Semigroup and Analyticity:** The semigroup $(T_t)_{t \ge 0}$ generated by $-(\Delta)^{\alpha/2}$ is strongly continuous, symmetric, and contractive on $L^2(\mathbb{R}^d)$. Moreover, it extends analytically to a sector in the complex plane:

$$ T_z f = \mathcal{F}^{-1}\left[e^{-z|\xi|^\alpha} \hat{f}(\xi)\right], \quad |\arg z| < \frac{\pi\alpha}{2}. \qquad (10.66) $$

The analyticity follows from the fact that the generator $-(\Delta)^{\alpha/2}$ is a sectorial operator on $L^2$, i.e.

$$ \|(\lambda I + (-\Delta)^{\alpha/2})^{-1}\|_{L^2 \to L^2} \le \frac{C}{|\lambda|}, \quad |\arg \lambda| < \pi - \theta_{\alpha}, \qquad (10.67) $$

for some sector angle $\theta_{\alpha} < \pi/2$. Hence, $(T_t)$ is analytic and contractive, satisfying

$$ \|T_t f\|_{L^2} \le \|f\|_{L^2}, \quad \forall t \ge 0. \qquad (10.68) $$

**Heat Kernel and Scaling Behavior:** The transition probability density (heat kernel) $p_t(x, y)$ of the symmetric $\alpha$-stable process satisfies the Fourier representation

$$ p_t(x, y) = \frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{i\xi \cdot (x-y)} e^{-t|\xi|^\alpha} d\xi. \qquad (10.69) $$

From this, one derives the fundamental two-sided heat kernel estimate:

$$ p_t(x, y) \asymp \frac{t}{(t^{1/\alpha} + |x-y|)^{d+\alpha}}, \qquad (10.70) $$

uniformly for all $t > 0$ and $x, y \in \mathbb{R}^d$. The notation $f(t, x, y) \asymp g(t, x, y)$ indicates the existence of constants $C_1, C_2 > 0$ such that

$$ C_1 g(t, x, y) \le f(t, x, y) \le C_2 g(t, x, y), \qquad (10.71) $$

for all arguments.

This estimate illustrates the heavy-tailed nature of jump propagation: whereas the classical Gaussian heat kernel decays exponentially in $|x - y|^2/t$, the fractional kernel decays polyno-
mially, allowing for long-range interactions. Consequently, the process generated by $-(\Delta)^{\alpha/2}$ exhibits nonlocal diffusion and infinite propagation speed, reflecting the essential difference between diffusive and jump-driven stochastic dynamics.

**Summary of Analytic and Spectral Properties:** In summary, for the fractional Laplacian and its general Lévy-type analogues:

• the operator is nonlocal, self-adjoint, and nonnegative on $L^2(\mathbb{R}^d)$;

• the spectrum is discrete under confinement potentials $V(x) \to +\infty$;

• eigenfunctions are Hölder continuous but not smooth;

• the semigroup is analytic and contractive on $L^2$;

• the heat kernel obeys heavy-tailed decay consistent with jump behavior.

These properties together reveal the deep interplay between spectral theory, regularity, and probabilistic structure in nonlocal analysis.

## 10.9 Conclusion

The extension from diffusive to jump processes represents a profound generalization of the classical probabilistic-analytic correspondence. While Brownian motion leads to local differential operators and continuous trajectories, Lévy processes and their Dirichlet forms give rise to nonlocal operators and càdlàg paths. This unified framework provides the foundation for modern analysis of stochastic systems with discontinuities, including anomalous diffusion, stable processes, and kinetic models, and forms a natural bridge between partial differential equations, harmonic analysis, and probability theory on infinite-dimensional spaces.

**References:** Applebaum (2009) [31], Sato (1999) [33], Bass et al. (2010) [34].
# Nonlocal PDEs and Fractional Diffusions

> “To understand diffusion beyond locality is to touch the geometry of space itself.”
> 
> — I. M. Gel’fand

## 11.1 Motivation and Overview

Fractional and nonlocal partial differential equations arise naturally as the infinitesimal descriptions of stochastic processes with jumps. While local diffusions correspond to second-order differential operators, nonlocal diffusions are governed by integro-differential operators whose behavior reflects the long-range interactions inherent in Lévy-type stochastic dynamics. Such equations have found central applications in anomalous transport, turbulence, quantitative finance, and nonlocal continuum mechanics.

The prototype of nonlocal diffusion is the fractional heat equation

$$ \partial_t u(t, x) + (-\Delta)^{\alpha/2} u(t, x) = 0, \quad (t, x) \in (0, \infty) \times \mathbb{R}^d, \qquad (11.1) $$

whose fundamental solution describes the transition density of an isotropic α-stable Lévy process. The fractional Laplacian here represents the generator of nonlocal diffusion, producing heavy-tailed propagation and infinite propagation speed in contrast to the Gaussian case.

## 11.2 Fractional Laplacian: Analytic Definitions

The fractional Laplacian $(-\Delta)^{\alpha/2}$ for $0 < \alpha < 2$ is a paradigmatic example of a nonlocal operator arising in analysis, probability, and geometry. It can be rigorously defined in several equivalent ways, each revealing different structural aspects of the operator — spectral, integral, and variational. In what follows, we present a mathematically rigorous derivation and justification of the three most fundamental formulations.

**Fourier Representation:** Let $f \in S(\mathbb{R}^d)$, the Schwartz space of rapidly decreasing smooth functions. The Fourier transform and its inverse are defined, respectively, as

$$ \hat{f}(\xi) = \mathcal{F}[f](\xi) = \int_{\mathbb{R}^d} e^{-ix \cdot \xi} f(x) dx, \quad f(x) = \frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{ix \cdot \xi} \hat{f}(\xi) d\xi. \qquad (11.2) $$
The fractional Laplacian is then defined spectrally by

$$ \mathcal{F}[-(\Delta)^{\alpha/2} f](\xi) = |\xi|^{\alpha} \hat{f}(\xi), \quad 0 < \alpha < 2. \qquad (11.3) $$

This identity is a natural extension of the classical Laplacian since for $\alpha = 2$,

$$ \mathcal{F}[-(\Delta)f](\xi) = |\xi|^2 \hat{f}(\xi). $$

Thus $(-\Delta)^{\alpha/2}$ can be viewed as a fractional power of the positive self-adjoint operator $-\Delta$ in the sense of the spectral theorem. Formally, if $-\Delta = \int_0^\infty \lambda dE(\lambda)$ denotes its spectral resolution, then

$$ (-\Delta)^{\alpha/2} = \int_{0}^{\infty} \lambda^{\alpha/2} dE(\lambda), \qquad (11.4) $$

which provides a rigorous spectral definition valid in $L^2(\mathbb{R}^d)$.

**Singular Integral Representation:** For functions $f \in C_c^\infty(\mathbb{R}^d)$, one can equivalently define the fractional Laplacian as a singular integral operator:

$$ (-\Delta)^{\alpha/2} f(x) = C_{d,\alpha} \text{P.V.} \int_{\mathbb{R}^d} \frac{f(x) - f(y)}{|x - y|^{d+\alpha}} dy, \quad 0 < \alpha < 2, \qquad (11.5) $$

where P.V. denotes the Cauchy principal value defined as

$$ \text{P.V.} \int_{\mathbb{R}^d} g(y) dy := \lim_{\epsilon \to 0} \int_{|y-x|>\epsilon} g(y) dy, \qquad (11.6) $$

and $C_{d,\alpha}$ is a dimension- and order-dependent normalization constant given explicitly by

$$ C_{d,\alpha} = \frac{2^{\alpha} \Gamma\left(\frac{d+\alpha}{2}\right)}{\pi^{d/2} |\Gamma(-\alpha/2)|}. \qquad (11.7) $$

This normalization ensures that the Fourier symbol of the operator in (11.5) equals $|\xi|^\alpha$, thereby making the integral and Fourier definitions equivalent. For each $f \in C_c^\infty(\mathbb{R}^d)$, the function

$$ K_{\alpha}(x-y) = \frac{C_{d,\alpha}}{|x-y|^{d+\alpha}} $$

defines a singular, non-integrable kernel at the origin. However, the antisymmetrization in the numerator $f(x) - f(y)$ renders the integral finite, since by Taylor expansion,

$$ f(x) - f(y) = \nabla f(x) \cdot (x - y) + O(|x - y|^2), $$

and hence the local singularity behaves like $|x - y|^{-d-\alpha+1}$, which is integrable near $y = x$ provided $\alpha < 2$. Thus the principal value integral exists pointwise for all smooth $f$.

This representation explicitly manifests the nonlocality of $(-\Delta)^{\alpha/2}$: the value at a point $x$ depends on all values of $f$ across $\mathbb{R}^d$, weighted by a kernel decaying like $|x - y|^{-d-\alpha}$. This property underlies the jump-diffusion character of the associated stochastic processes.
**Dirichlet Form Representation:** Let $H^{\alpha/2}(\mathbb{R}^d)$ denote the fractional Sobolev space, defined as

$$H^{\alpha/2}(\mathbb{R}^d) = \left\{ f \in L^2(\mathbb{R}^d) : \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy < \infty \right\}. \quad (11.8)$$

On this space, we define the bilinear form

$$\mathcal{E}(f,g) = \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(f(x) - f(y))(\overline{g(x)} - \overline{g(y)})}{|x - y|^{d+\alpha}} dx dy, \quad (11.9)$$

and the corresponding energy

$$\mathcal{E}(f, f) = \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy. \quad (11.10)$$

The form $\mathcal{E}$ is:

* **Symmetric:** $\mathcal{E}(f,g) = \overline{\mathcal{E}(g,f)}$.

* **Closed:** The domain $H^{\alpha/2}(\mathbb{R}^d)$ is complete under the norm

$$\|f\|_{H^{\alpha/2}}^2 = \|f\|_{L^2}^2 + \mathcal{E}(f,f).$$

* **Markovian:** For any normal contraction $\eta: \mathbb{R} \to \mathbb{R}$ with $|\eta(a) - \eta(b)| \le |a - b|$, we have

$$\mathcal{E}(\eta \circ f, \eta \circ f) \le \mathcal{E}(f, f),$$

ensuring that the corresponding semigroup preserves positivity and contractivity.

By the theory of symmetric Dirichlet forms (Fukushima-Oshima-Takeda), the form $\mathcal{E}$ uniquely determines a self-adjoint, nonnegative operator $A$ on $L^2(\mathbb{R}^d)$ satisfying

$$\mathcal{E}(f,g) = \langle Af,g \rangle_{L^2}, \quad f \in \mathcal{D}(A), g \in H^{\alpha/2}(\mathbb{R}^d), \quad (11.11)$$

and this operator coincides with the fractional Laplacian $A = (-\Delta)^{\alpha/2}$. Hence, the three formulations — Fourier, singular integral, and Dirichlet form — are rigorously equivalent on their common domains.

**Summary:**

<table><thead><tr><td>Definition Type</td><td>Formula</td><td>Key Feature</td></tr></thead><tbody><tr><td>Fourier</td><td>$\mathcal{F}[-(\Delta)^{\alpha/2}f](\xi) = |\xi|^{\alpha} \hat{f}(\xi)$</td><td>Spectral structure</td></tr><tr><td>Singular Integral</td><td>$(\Delta)^{\alpha/2}f(x) = C_{d,\alpha} \text{P.V.} \int \frac{f(x) - f(y)}{|x - y|^{d+\alpha}} dy$</td><td>Nonlocality</td></tr><tr><td>Dirichlet Form</td><td>$\mathcal{E}(f,f) = \frac{C_{d,\alpha}}{2} \iint \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy$</td><td>Variational energy</td></tr></tbody></table>

Each perspective exposes a different analytic facet of the fractional Laplacian — its spectral nature, its nonlocal integral structure, and its variational interpretation as the generator of a symmetric $\alpha$-stable Lévy process.
## 11.3 Fractional Heat Equation and Its Kernel

The fractional heat equation is the evolution equation governed by the nonlocal operator $(-\Delta)^{\alpha/2}$ for $0 < \alpha < 2$. Formally, it reads

$$ \partial_t u(t, x) + (-\Delta)^{\alpha/2} u(t, x) = 0, \quad t > 0, x \in \mathbb{R}^d, \qquad (11.12) $$

with the initial condition $u(0, x) = f(x) \in L^2(\mathbb{R}^d)$. The solution defines a strongly continuous semigroup $(P_t)_{t \ge 0}$ on $L^2(\mathbb{R}^d)$, given by

$$ P_t f = e^{-t(-\Delta)^{\alpha/2}} f. \qquad (11.13) $$

**Fourier Representation of the Semigroup:** By the spectral characterization of the fractional Laplacian, we have

$$ \mathcal{F}[(-\Delta)^{\alpha/2} f](\xi) = |\xi|^\alpha \hat{f}(\xi), $$

which immediately implies that the semigroup acts diagonally in Fourier space:

$$ \mathcal{F}[P_t f](\xi) = e^{-t|\xi|^\alpha} \hat{f}(\xi), \quad t \ge 0, \xi \in \mathbb{R}^d. \qquad (11.14) $$

Taking the inverse Fourier transform yields the convolution representation

$$ P_t f(x) = (p_t * f)(x), \quad \text{where } p_t(x) = \frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{ix \cdot \xi} e^{-t|\xi|^\alpha} d\xi. \qquad (11.15) $$

The function $p_t(x)$ serves as the fractional heat kernel, or equivalently, the transition density of the isotropic symmetric $\alpha$-stable Lévy process.

**Self-Similar Scaling Property:** A fundamental structural property of $p_t(x)$ is its self-similarity (or scaling invariance). By performing a change of variables $\xi' = t^{1/\alpha}\xi$ in (11.15), one obtains

$$ \begin{aligned} p_t(x) &= \frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{ix \cdot \xi} e^{-t|\xi|^\alpha} d\xi = \frac{t^{-d/\alpha}}{(2\pi)^d} \int_{\mathbb{R}^d} e^{i(t^{-1/\alpha}x) \cdot \xi'} e^{-|\xi'|^\alpha} d\xi' \\ &= t^{-d/\alpha} p_1(t^{-1/\alpha}x). \end{aligned} \qquad (11.16) $$

This homogeneity relation reflects the scaling behavior of the generator:

$$ (-\Delta)^{\alpha/2}(f(\lambda \cdot))(x) = \lambda^{\alpha} ((-\Delta)^{\alpha/2} f)(\lambda x), $$

showing that the process is self-similar with index $1/\alpha$.

**Probabilistic Interpretation:** The fractional heat semigroup admits the probabilistic representation

$$ (P_t f)(x) = \mathbb{E}^x[f(X_t)], \qquad (11.17) $$

where $(X_t)_{t \ge 0}$ is an isotropic symmetric $\alpha$-stable Lévy process with characteristic function

$$ \mathbb{E}^0[e^{i\xi \cdot X_t}] = e^{-t|\xi|^\alpha}. $$
Thus, $p_t(x)$ is the probability density function of $X_t$, i.e.,

$$p_t(x) dx = \mathbb{P}(X_t \in dx).$$

In particular, the process exhibits jumps with heavy-tailed increments, in contrast to the Gaussian increments of Brownian motion.

**Heat Kernel Estimates:** The kernel $p_t(x)$ is positive, radially symmetric, and satisfies the following precise asymptotic estimates: there exist constants $c_1, c_2 > 0$, depending only on $d$ and $\alpha$, such that for all $t > 0$ and $x \in \mathbb{R}^d$,

$$c_1 \frac{t}{(t^{1/\alpha} + |x|)^{d+\alpha}} \le p_t(x) \le c_2 \frac{t}{(t^{1/\alpha} + |x|)^{d+\alpha}}. \quad (11.18)$$

The proof of these inequalities rests on detailed asymptotic analysis of the Fourier integral and Tauberian theorems relating the decay of $e^{-t|\xi|^\alpha}$ to the behavior of $p_t(x)$ at infinity.

**Asymptotic Behavior:** From (11.18), we infer:

* For small displacements ($|x| \ll t^{1/\alpha}$):

$$p_t(x) \approx c t^{-d/\alpha},$$

which matches the density scaling of a stable process.

* For large displacements ($|x| \gg t^{1/\alpha}$):

$$p_t(x) \approx c' \frac{t}{|x|^{d+\alpha}},$$

showing a power-law, heavy-tailed decay unlike the Gaussian exponential decay of the classical heat kernel.

**Comparison with Gaussian Heat Kernel:** For the classical Laplacian ($\alpha = 2$), the heat kernel is

$$p_t^{(2)}(x) = (4\pi t)^{-d/2} e^{-\frac{|x|^2}{4t}},$$

which exhibits exponential decay in $|x|$. In contrast, for $0 < \alpha < 2$, the fractional heat kernel $p_t(x)$ decays polynomially, encoding the presence of long-range jumps and Lévy flight behavior. This difference marks the transition from local diffusion to nonlocal anomalous diffusion.

**Summary:** The fractional heat semigroup $(P_t)_{t \ge 0}$ generated by $-(\Delta)^{\alpha/2}$ satisfies:

$$\mathcal{F}[p_t](\xi) = e^{-t|\xi|^\alpha}, \quad (11.19)$$

$$p_t(x) = t^{-d/\alpha} p_1(t^{-1/\alpha}x), \quad (11.20)$$

$$c_1 \frac{t}{(t^{1/\alpha} + |x|)^{d+\alpha}} \le p_t(x) \le c_2 \frac{t}{(t^{1/\alpha} + |x|)^{d+\alpha}}. \quad (11.21)$$

These results rigorously characterize the analytic and probabilistic structure of the fractional heat flow, revealing its self-similar scaling, nonlocal propagation, and heavy-tailed behavior intrinsic to stable Lévy processes.
## 11.4 Cauchy Problems and Well-Posedness

We now provide a mathematically rigorous treatment of the Cauchy problem associated with the fractional Laplacian. Let $0 < \alpha < 2$ and consider the equation

$$ \begin{cases} \partial_t u(t, x) + (-\Delta)^{\alpha/2} u(t, x) = 0, & t > 0, x \in \mathbb{R}^d, \\ u(0, x) = u_0(x), & x \in \mathbb{R}^d. \end{cases} \qquad (11.22) $$

We seek to establish existence, uniqueness, and continuous dependence of solutions in an appropriate functional setting.

**Functional Analytic Framework:** Let $L^2(\mathbb{R}^d)$ denote the Hilbert space with the standard inner product

$$ \langle f, g \rangle = \int_{\mathbb{R}^d} f(x) \overline{g(x)} dx, $$

and norm $||f||_{L^2} = \langle f, f \rangle^{1/2}$. Define the operator $A := (-\Delta)^{\alpha/2}$ on the domain

$$ \mathcal{D}(A) = \{ f \in L^2(\mathbb{R}^d) : |\xi|^{\alpha} \hat{f}(\xi) \in L^2(\mathbb{R}^d) \}, $$

that is, $\mathcal{D}(A) = H^\alpha(\mathbb{R}^d)$, the fractional Sobolev space.

For $f \in \mathcal{D}(A)$, we have

$$ \|Af\|_{L^2}^2 = \int_{\mathbb{R}^d} |\xi|^{2\alpha} |\hat{f}(\xi)|^2 d\xi < \infty. $$

The operator $A$ is self-adjoint, positive definite, and densely defined. Therefore, by the spectral theorem, it generates a strongly continuous contraction semigroup $\{P_t\}_{t \ge 0}$ on $L^2(\mathbb{R}^d)$ given by

$$ P_t = e^{-tA}. $$

**Mild and Weak Formulations:** The mild solution to (11.22) is defined by

$$ u(t) = P_t u_0 = e^{-tA} u_0. \qquad (11.23) $$

In the Fourier domain, this is explicitly given by

$$ \hat{u}(t, \xi) = e^{-t|\xi|^\alpha} \hat{u}_0(\xi), $$

which, by Plancherel's theorem, ensures that $u(t) \in L^2(\mathbb{R}^d)$ for all $t \ge 0$. Moreover, the strong continuity of $(P_t)$ implies

$$ \lim_{t \to 0^+} \|P_t u_0 - u_0\|_{L^2} = 0, $$

so that $u \in C([0, \infty); L^2(\mathbb{R}^d))$. A function $u : [0, \infty) \to L^2(\mathbb{R}^d)$ is called a weak solution of (11.22) if, for all $\varphi \in \mathcal{D}(A)$,

$$ \frac{d}{dt} \langle u(t), \varphi \rangle + \langle A^{1/2} u(t), A^{1/2} \varphi \rangle = 0, \quad \text{for almost every } t > 0, $$
and $u(0) = u_0$. By the semigroup representation (11.23), the mild and weak solutions coincide.

**Contractivity and Energy Dissipation:** The semigroup $(P_t)_{t \ge 0}$ satisfies the following fundamental properties:

* (Strong continuity): $\lim_{t \to 0^+} \|P_t f - f\|_{L^2} = 0$ for all $f \in L^2(\mathbb{R}^d)$.

* (Self-adjointness): $\langle P_t f, g \rangle = \langle f, P_t g \rangle$ for all $f, g \in L^2(\mathbb{R}^d)$.

* (Contractivity): $\|P_t f\|_{L^2} \le \|f\|_{L^2}$ for all $t \ge 0$.

Differentiating the $L^2$-norm along the solution $u(t)$ yields the energy dissipation identity

$$ \frac{d}{dt} \|u(t)\|_{L^2}^2 = -2 \langle A^{1/2} u(t), A^{1/2} u(t) \rangle = -2 \|A^{1/2} u(t)\|_{L^2}^2 \le 0. \quad (11.24) $$

Thus, the $L^2$-norm of the solution is non-increasing in time:

$$ \|u(t)\|_{L^2} \le \|u_0\|_{L^2}, \quad \forall t \ge 0. $$

**Uniqueness and Well-Posedness:** To show uniqueness, suppose $u_1$ and $u_2$ are two mild solutions with the same initial data. Then $w = u_1 - u_2$ satisfies

$$ \partial_t w + Aw = 0, \quad w(0) = 0. $$

Taking the inner product with $w(t)$ and integrating over $\mathbb{R}^d$, we find

$$ \frac{1}{2} \frac{d}{dt} \|w(t)\|_{L^2}^2 + \|A^{1/2} w(t)\|_{L^2}^2 = 0. $$

Since $\|w(0)\|_{L^2} = 0$, Grönwall's inequality gives $\|w(t)\|_{L^2} = 0$ for all $t$, hence $u_1 = u_2$. Therefore, the solution is unique in $C([0, \infty); L^2(\mathbb{R}^d))$.

**Conclusion:** The Cauchy problem (11.22) admits a unique mild (equivalently, weak) solution

$$ u(t, x) = (P_t u_0)(x) = (p_t * u_0)(x), $$

with $p_t$ the fractional heat kernel. The mapping $u_0 \mapsto u(t)$ defines a strongly continuous, symmetric, and contractive semigroup on $L^2(\mathbb{R}^d)$. Hence, the fractional heat equation is *well-posed* in the $L^2$ sense, with continuous dependence on the initial data and conservation (or decay) of the $L^2$ energy.

## 11.5 Fractional Schrödinger Operators

We now construct and analyze the fractional Schrödinger operator, which generalizes the classical Schrödinger operator to incorporate nonlocal effects governed by the fractional Laplacian.

**Definition and Functional Setting:** Let $0 < \alpha < 2$. The fractional Schrödinger operator is defined formally as

$$ H = (-\Delta)^{\alpha/2} + V(x), \quad (11.25) $$
where $V: \mathbb{R}^d \to \mathbb{R}$ is a measurable potential function. The operator acts on the Hilbert space $L^2(\mathbb{R}^d)$. The natural domain of $H$ is determined by the intersection of the domains of its two components:

$$ \mathcal{D}(H) = \{ f \in H^{\alpha}(\mathbb{R}^d) : V(x)f(x) \in L^2(\mathbb{R}^d) \}. $$

Thus, for $f \in \mathcal{D}(H)$, we define

$$ Hf = (-\Delta)^{\alpha/2} f + V(x)f(x), $$

where $(-\Delta)^{\alpha/2}$ is the fractional Laplacian understood in the Fourier sense,

$$ (\widehat{-\Delta})^{\alpha/2} f(\xi) = |\xi|^{\alpha} \hat{f}(\xi). $$

**Quadratic form representation:** Define the quadratic form associated with $H$ by

$$ \mathcal{E}_V(f,g) = \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(f(x) - f(y))(g(x) - g(y))}{|x - y|^{d+\alpha}} dx dy + \int_{\mathbb{R}^d} V(x)f(x)g(x) dx, \quad (11.26) $$

for $f, g \in C_c^\infty(\mathbb{R}^d)$, where $C_{d,\alpha}$ is the normalization constant of the fractional Laplacian. The form $\mathcal{E}_V$ is closable on $L^2(\mathbb{R}^d)$ whenever $V$ belongs to the Kato class $K_{d,\alpha}$, which ensures suitable integrability and smallness properties of the potential.

**Self-Adjointness and Semiboundedness:** The quadratic form (11.26) is densely defined, symmetric, and semibounded from below if

$$ \inf_{x \in \mathbb{R}^d} V(x) > -\infty. $$

By the Kato-Friedrichs representation theorem, there exists a unique self-adjoint operator $H$ on $L^2(\mathbb{R}^d)$ such that

$$ \mathcal{E}_V(f,g) = \langle H^{1/2}f, H^{1/2}g \rangle, \quad f,g \in \mathcal{D}(\mathcal{E}_V), $$

and this operator coincides with (11.25) in the distributional sense. Moreover, the spectrum $\sigma(H) \subseteq [\inf V, \infty)$ is real and bounded from below.

**Feynman-Kac Representation for the Semigroup:** Let $(X_t)_{t \ge 0}$ denote a symmetric $\alpha$-stable Lévy process on $\mathbb{R}^d$, characterized by the Fourier transform

$$ \mathbb{E}^0[e^{i\langle\xi, X_t\rangle}] = e^{-t|\xi|^\alpha}. $$

The generator of $(X_t)$ is $-(\Delta)^{\alpha/2}$. For $V$ in the Kato class, the semigroup $\{e^{-tH}\}_{t \ge 0}$ associated with $H$ admits a probabilistic representation known as the *nonlocal Feynman-Kac formula*:

$$ (e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp\left(-\int_0^t V(X_s) ds \right) f(X_t) \right], \quad (11.27) $$

where $\mathbb{E}^x$ denotes the expectation for the process started at $X_0 = x$.

For the justification of (11.27), the identity (11.27) follows from the following steps:
1. The semigroup $(P_t)_{t \ge 0}$ generated by $-(-\Delta)^{\alpha/2}$ acts as

$$ (P_t f)(x) = \mathbb{E}^x[f(X_t)] = (p_t * f)(x), $$

with Fourier symbol $e^{-t|\xi|^\alpha}$.

2. Define the multiplicative functional

$$ M_t = \exp\left(-\int_0^t V(X_s) ds\right), $$

which is positive and satisfies $M_t \le e^{t\|V_-\|\infty}$ if $V_-$ denotes the negative part of $V$.

3. Then, the family $\{T_t\}_{t \ge 0}$ defined by

$$ (T_t f)(x) = \mathbb{E}^x[M_t f(X_t)] $$

forms a strongly continuous, symmetric, positivity-preserving semigroup on $L^2(\mathbb{R}^d)$.

4. The generator of $(T_t)$ is the self-adjoint operator $H$ defined in (11.25).

**Analytic and Probabilistic Equivalence:** The representation (11.27) establishes an exact analytic-probabilistic equivalence:

$$ e^{-tH} = \mathbb{E}\left[\exp\left(-\int_{0}^{t} V(X_s) ds\right) \cdot f(X_t)\right], $$

linking the semigroup generated by the nonlocal operator $H$ with expectations over the paths of the Lévy process $(X_t)$.

This duality provides a powerful bridge between spectral theory and stochastic processes:

* The analytic properties of $H$ (self-adjointness, spectral bounds, domain structure) correspond to path properties of $(X_t)$ and integrability of the exponential functional of $V$.

* The lower spectral bound of $H$ satisfies

$$ \inf \sigma(H) \ge \inf_{x \in \mathbb{R}^d} V(x), $$

reflecting the fact that the potential term shifts the energy spectrum upward by its infi-mum.

**Summary:** Under the assumption that $V$ belongs to the Kato class $K_{d,\alpha}$ and is bounded from below, the fractional Schrödinger operator

$$ H = (-\Delta)^{\alpha/2} + V(x) $$

is self-adjoint, semibounded, and generates a strongly continuous symmetric semigroup $(e^{-tH})_{t \ge 0}$ on $L^2(\mathbb{R}^d)$ satisfying the nonlocal Feynman-Kac representation (11.27). The spectrum of $H$ is purely real and satisfies

$$ \sigma(H) \subseteq [\inf V, \infty), $$
demonstrating that the analytic and probabilistic formulations of the nonlocal quantum evolution are fully consistent.

## 11.6 Boundary Value Problems and Extension Technique

A remarkable and deep connection between nonlocal and local partial differential equations was discovered by Caffarelli and Silvestre, showing that the fractional Laplacian can be realized as a Dirichlet-to-Neumann map for a degenerate elliptic problem posed in one higher dimension. This correspondence allows one to translate nonlocal problems into local PDEs with weighted ellipticity, thereby enabling the use of classical analytic tools in nonlocal analysis.

**The Extension Problem:** Let $\alpha \in (0,2)$ and define the extended domain as the upper half-space

$$ \mathbb{R}_+^{d+1} = \{(x, y) \in \mathbb{R}^d \times (0, \infty)\}. $$

Given boundary data $f: \mathbb{R}^d \to \mathbb{R}$, consider the weighted elliptic boundary value problem

$$ \left\{ \begin{array}{ll} \nabla \cdot (y^{1-\alpha} \nabla U(x, y)) = 0, & (x, y) \in \mathbb{R}_+^{d+1}, \\ U(x, 0) = f(x), & x \in \mathbb{R}^d. \end{array} \right. \qquad (11.28) $$

The function $U$ is called the $\alpha$-harmonic extension of $f$. The equation in (11.28) is a degenerate elliptic PDE, since the coefficient $y^{1-\alpha}$ vanishes or blows up near $y=0$ depending on whether $\alpha < 1$ or $\alpha > 1$.

The operator

$$ L_{\alpha}U := \nabla \cdot (y^{1-\alpha}\nabla U) $$

is self-adjoint on the weighted space $L^2(\mathbb{R}_+^{d+1}, y^{1-\alpha} dx dy)$, and the corresponding weak formulation reads:

$$ \int_{\mathbb{R}_+^{d+1}} y^{1-\alpha} \nabla U \cdot \nabla \varphi dx dy = 0, \quad \forall \varphi \in C_c^{\infty}(\mathbb{R}_+^{d+1}). $$

**Poisson Kernel Representation:** The solution $U$ to (11.28) can be represented explicitly by convolution with a suitable Poisson kernel:

$$ U(x, y) = (P_y^{(\alpha)} * f)(x), \qquad (11.29) $$

where the $\alpha$-Poisson kernel is given by

$$ P_y^{(\alpha)}(x) = c_{d,\alpha} \frac{y^{\alpha}}{(|x|^2 + y^2)^{\frac{d+\alpha}{2}}}, \quad c_{d,\alpha} = \frac{\Gamma(\frac{d+\alpha}{2})}{\pi^{d/2} \Gamma(\frac{\alpha}{2})}. \qquad (11.30) $$

This kernel satisfies $\int_{\mathbb{R}^d} P_y^{(\alpha)}(x) dx = 1$ for all $y > 0$, and one easily checks that $L_{\alpha}U = 0$ in $\mathbb{R}_+^{d+1}$ and $\lim_{y \to 0^+} U(x, y) = f(x)$.

**Dirichlet-to-Neumann Map and the Fractional Laplacian:** The key identity discovered
by Caffarelli and Silvestre is the correspondence

$$(-\Delta)^{\alpha/2} f(x) = -C_{\alpha} \lim_{y \to 0^+} y^{1-\alpha} \partial_y U(x, y), \quad (11.31)$$

where $U$ solves (11.28) and

$$C_{\alpha} = \frac{2^{\alpha-1}\Gamma(\alpha/2)}{\Gamma(1-\alpha/2)}.$$

This identity expresses the fractional Laplacian as a boundary flux associated with the weighted harmonic extension of $f$. The right-hand side is well-defined for $f \in H^{\alpha/2}(\mathbb{R}^d)$, and the limit exists in the sense of distributions.

**Variational Formulation and Energy Identity:** The extension problem admits a natural variational structure. Define the weighted energy functional

$$\mathcal{E}_{\alpha}(U) = \frac{1}{2} \int_{\mathbb{R}_{+}^{d+1}} y^{1-\alpha} |\nabla U(x, y)|^2 dx dy. \quad (11.32)$$

Among all functions $U$ with prescribed boundary trace $U(\cdot, 0) = f$, the minimizer of $\mathcal{E}_{\alpha}$ satisfies the Euler-Lagrange equation $L_{\alpha}U = 0$, i.e., it solves the extension problem (11.28).

Moreover, there is an exact equivalence of energies:

$$\mathcal{E}_{\alpha}(U) = C_{d,\alpha} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|f(x) - f(y)|^2}{|x - y|^{d+\alpha}} dx dy = 2C_{d,\alpha} \mathcal{E}(f, f), \quad (11.33)$$

where $\mathcal{E}(f, f)$ is the Dirichlet form of the fractional Laplacian. This shows that minimizing the weighted local energy $\mathcal{E}_{\alpha}(U)$ in $\mathbb{R}_{+}^{d+1}$ is equivalent to minimizing the nonlocal energy on $\mathbb{R}^d$.

**Analytic Consequences:** The representation (11.31) has several fundamental consequences:

* **Boundary regularity:** Solutions to fractional elliptic equations $(-\Delta)^{\alpha/2}f = g$ inherit regularity from the corresponding degenerate elliptic equation $L_{\alpha}U = 0$.

* **Spectral interpretation:** On bounded domains $\Omega \subset \mathbb{R}^d$, the spectral fractional Laplacian defined via eigenfunction expansions coincides with the Dirichlet-to-Neumann map of the extension problem with vanishing boundary condition $U|_{\partial\Omega \times (0, \infty)} = 0$.

* **Variational analysis:** The correspondence (11.33) permits the application of tools from weighted Sobolev spaces and calculus of variations to fractional problems.

* **Harnack and Liouville principles:** The weighted harmonic nature of $U$ yields nonlocal analogues of classical results such as Harnack inequalities and Liouville theorems for $\alpha$-harmonic functions.

**Summary:** The Caffarelli-Silvestre extension provides a rigorous geometric and analytic realization of the fractional Laplacian as a boundary flux:

$$(-\Delta)^{\alpha/2} f(x) = -C_{\alpha} \lim_{y \to 0^+} y^{1-\alpha} \partial_y U(x, y),$$
where $U$ solves the degenerate elliptic problem $\nabla \cdot (y^{1-\alpha}\nabla U) = 0$. This construction converts nonlocal problems on $\mathbb{R}^d$ into local PDEs on $\mathbb{R}_+^{d+1}$, thereby bridging fractional analysis, variational methods, and geometric PDE theory in a unified framework.

## 11.7 Nonlocal Variational Problems and Energy Minimization

In the mathematical theory of nonlocal partial differential equations, a central principle is that weak or distributional solutions can often be interpreted as critical points or minimizers of appropriately defined nonlocal energy functionals. This variational viewpoint extends the classical Dirichlet principle to the nonlocal framework governed by fractional Laplacians.

Consider the fractional Poisson equation

$$(-\Delta)^{\alpha/2} u = f \quad \text{in } \mathbb{R}^d, \qquad (11.34)$$

where $0 < \alpha < 2$ and $f \in H^{-\alpha/2}(\mathbb{R}^d)$ is a given distribution. The operator $(-\Delta)^{\alpha/2}$ is nonlocal and self-adjoint on $L^2(\mathbb{R}^d)$, with domain $H^{\alpha/2}(\mathbb{R}^d)$. Its action may be represented weakly as

$$\langle (-\Delta)^{\alpha/2} u, v \rangle = C_{d,\alpha} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(u(x) - u(y))(v(x) - v(y))}{|x - y|^{d+\alpha}} dx dy, \qquad (11.35)$$

for all $u, v \in H^{\alpha/2}(\mathbb{R}^d)$, where $C_{d,\alpha}$ is a normalization constant depending only on $d$ and $\alpha$.

The weak formulation of the fractional Poisson equation seeks $u \in H^{\alpha/2}(\mathbb{R}^d)$ such that

$$\mathcal{E}(u, v) = \langle f, v \rangle, \quad \forall v \in H^{\alpha/2}(\mathbb{R}^d), \qquad (11.36)$$

where the bilinear form $\mathcal{E}(\cdot, \cdot)$ is defined by

$$\mathcal{E}(u, v) := \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(u(x) - u(y))(v(x) - v(y))}{|x - y|^{d+\alpha}} dx dy. \qquad (11.37)$$

This bilinear form is symmetric, coercive, and continuous on $H^{\alpha/2}(\mathbb{R}^d)$, that is,

$$\mathcal{E}(u, u) \geq c_{\alpha,d} \|u\|_{H^{\alpha/2}(\mathbb{R}^d)}^2, \quad |\mathcal{E}(u, v)| \leq C_{\alpha,d} \|u\|_{H^{\alpha/2}} \|v\|_{H^{\alpha/2}},$$

for positive constants $c_{\alpha,d}$, $C_{\alpha,d}$. By the Lax-Milgram theorem, there exists a unique weak solution $u$ satisfying this equation.

Equivalently, $u$ minimizes the nonlocal energy functional

$$\mathcal{J}(u) = \frac{C_{d,\alpha}}{4} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|u(x) - u(y)|^2}{|x - y|^{d+\alpha}} dx dy - \int_{\mathbb{R}^d} f(x)u(x) dx. \qquad (11.38)$$

To see this, one observes that for any perturbation $u + \varepsilon v$ with $v \in H^{\alpha/2}(\mathbb{R}^d)$, the first variation of $\mathcal{J}$ satisfies

$$\begin{aligned} \frac{d}{d\varepsilon} \mathcal{J}(u + \varepsilon v) \Big|_{\varepsilon=0} &= \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(u(x) - u(y))(v(x) - v(y))}{|x - y|^{d+\alpha}} dx dy - \int_{\mathbb{R}^d} f(x)v(x) dx \\ &= \mathcal{E}(u, v) - \langle f, v \rangle. \end{aligned} \qquad (11.39)$$
Setting this to zero for all admissible $v$ yields precisely the weak form of the fractional Poisson problem. Thus, the variational minimizer of $\mathcal{J}$ is the weak solution to $(-\Delta)^{\alpha/2}u = f$.

The energy $\mathcal{E}(u, u)$ admits a Fourier-space representation:

$$ \mathcal{E}(u, u) = c_{d,\alpha} \int_{\mathbb{R}^d} |\xi|^\alpha |\hat{u}(\xi)|^2 d\xi, \qquad (11.40) $$

establishing the equivalence between the Gagliardo seminorm and the spectral norm of $H^{\alpha/2}$. This characterization emphasizes that the fractional Laplacian penalizes high-frequency oscillations according to a power-law weight, capturing nonlocal smoothness properties intrinsic to fractional-order Sobolev spaces.

Therefore, the fractional Poisson equation is well-posed in $H^{\alpha/2}(\mathbb{R}^d)$: there exists a unique weak solution depending continuously on $f$, and the solution minimizes the nonlocal Dirichlet energy subject to the forcing term. This framework generalizes the classical elliptic variational principle to encompass long-range interactions governed by power-law kernels.

## 11.8 Spectral Properties and Regularity

The spectral structure of the fractional Laplacian on bounded domains provides a rigorous nonlocal analogue of the classical elliptic spectral theory. Let $\Omega \subset \mathbb{R}^d$ be a bounded open domain with sufficiently smooth boundary, and let $0 < \alpha < 2$. We define the Dirichlet fractional Laplacian $(-\Delta)^{\alpha/2}$ as the self-adjoint operator associated with the quadratic form

$$ \mathcal{E}(u, v) = \frac{C_{d,\alpha}}{2} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{(u(x) - u(y))(v(x) - v(y))}{|x - y|^{d+\alpha}} dx dy, \qquad (11.41) $$

with domain

$$ \mathcal{D}(\mathcal{E}) = \{u \in H^{\alpha/2}(\mathbb{R}^d) : u = 0 \text{ a.e. in } \mathbb{R}^d \setminus \Omega\}. $$

This definition enforces a nonlocal Dirichlet condition by penalizing jumps across the boundary. The operator $(-\Delta)^{\alpha/2}$ acts as the generator of a symmetric Markov semigroup $(P_t)_{t \ge 0}$ on $L^2(\Omega)$, defined by $P_t = e^{-t(-\Delta)^{\alpha/2}}$.

**Spectral decomposition:** Since $\mathcal{E}$ is a symmetric, densely defined, closed, and coercive bilinear form on $L^2(\Omega)$, the associated operator $(-\Delta)^{\alpha/2}$ is self-adjoint, positive, and possesses a compact resolvent. Consequently, by the spectral theorem for compact self-adjoint operators, there exists an orthonormal basis of eigenfunctions

$$ \{\phi_k\}_{k=1}^{\infty} \subset L^2(\Omega) $$

and an increasing sequence of eigenvalues

$$ 0 < \lambda_1 < \lambda_2 \le \lambda_3 \le \dots, \quad \lambda_k \to +\infty, $$

such that

$$ (-\Delta)^{\alpha/2} \phi_k = \lambda_k \phi_k, \quad \text{with } \phi_k|_{\mathbb{R}^d \setminus \Omega} = 0. \qquad (11.42) $$
The eigenfunctions form a complete orthonormal basis of $L^2(\Omega)$, that is,

$$ \langle \phi_j, \phi_k \rangle_{L^2(\Omega)} = \delta_{jk}, \quad f = \sum_{k=1}^{\infty} \langle f, \phi_k \rangle \phi_k, \quad \forall f \in L^2(\Omega). $$

**Semigroup representation:** The semigroup $(P_t)$ generated by $-(\Delta)^{\alpha/2}$ admits the spectral expansion

$$ (P_t f)(x) = \sum_{k=1}^{\infty} e^{-\lambda_k t} \langle f, \phi_k \rangle \phi_k(x), \quad (11.43) $$

which converges in $L^2(\Omega)$ for each $t > 0$. This representation provides an explicit solution formula for the fractional heat equation

$$ \partial_t u + (-\Delta)^{\alpha/2} u = 0, \quad u(0, \cdot) = f, $$

given by $u(t) = P_t f$.

**Spectral asymptotics:** The eigenvalues satisfy a Weyl-type asymptotic law analogous to that for the classical Laplacian:

$$ \lambda_k \sim C_{d,\alpha} \left( \frac{k}{|\Omega|} \right)^{\alpha/d}, \quad \text{as } k \to \infty, \quad (11.44) $$

where $C_{d,\alpha}$ depends only on the dimension and the fractional order. This relation quantifies the scaling of high-frequency eigenmodes and encodes the nonlocal dispersion inherent in the operator.

**Regularity of eigenfunctions:** Unlike the classical Laplacian, the fractional Laplacian lacks elliptic locality; consequently, the eigenfunctions $\phi_k$ are generally not smooth up to the boundary. For bounded $\Omega$ with Lipschitz boundary, one has

$$ \phi_k \in C^\beta(\bar{\Omega}), \quad \beta < \alpha, $$

and this Hölder regularity is optimal in general. Interior regularity is stronger: for $\Omega' \subset \Omega$, $\phi_k \in C^\infty(\Omega')$, reflecting analytic smoothing in the interior but nonlocal singularity propagation near $\partial\Omega$. These results stem from the nonlocal elliptic estimates of Silvestre and Ros-Oton-Serra, which show that $(-\Delta)^{\alpha/2}$ gains only fractional derivatives in the boundary setting.

**Functional analytic interpretation:** The fractional Laplacian defines an unbounded self-adjoint operator on $L^2(\Omega)$ with domain

$$ \mathcal{D}((-\Delta)^{\alpha/2}) = \{u \in H_0^{\alpha/2}(\Omega) : (-\Delta)^{\alpha/2} u \in L^2(\Omega)\}. $$

The semigroup $(P_t)$ is analytic and contractive on $L^2(\Omega)$, and satisfies

$$ \|P_t f\|_{L^2} \le \|f\|_{L^2}, \quad \frac{d}{dt} \|P_t f\|_{L^2}^2 = -2\mathcal{E}(P_t f, P_t f) \le 0. $$

Thus, the fractional heat flow preserves $L^2$ norm and dissipates the nonlocal Dirichlet energy, reflecting both spectral decay and probabilistic contractivity of the corresponding $\alpha$-stable process.
In summary, the spectral theory of the fractional Laplacian retains the orthogonal completeness and semigroup representation of classical elliptic operators, but with markedly different regularity characteristics: smoothness is lost at the boundary, replaced by fractional Hölder continuity that reflects the intrinsic nonlocality of the underlying operator.

## 11.9 Conclusion

Fractional and nonlocal partial differential equations extend the classical theory of diffusion to encompass long-range interactions and anomalous transport. Their deep connection with Lévy processes, Dirichlet forms, and spectral theory reveals a unified structure binding probability, functional analysis, and geometry. Through the Feynman-Kac correspondence, every nonlocal evolution acquires both analytic and stochastic meaning — establishing a bridge between PDEs, random jumps, and the geometry of nonlocality.

**References:** Di Nezza et al. 2012 [36], Servadei and Valdinoci 2015 [37].
# Infinite-Dimensional Diffusions and Stochastic Partial Differential Equations

> “Probability in infinite dimensions is not mere chance—it is geometry itself.”
> 
> — S. R. S. Varadhan

## 12.1 Motivation and Overview

Infinite-dimensional diffusions arise naturally when the state of a stochastic system is described by a random field rather than a finite-dimensional vector. Such systems include stochastic evolution of temperature fields, fluid velocities, concentration profiles, or quantum states, each governed by partial differential equations perturbed by random noise. The rigorous analytic framework for their study combines infinite-dimensional analysis, functional analysis on Hilbert and Banach spaces, and stochastic calculus in infinite dimensions.

A stochastic partial differential equation (SPDE) in abstract form is typically written as

$$dX_t = (AX_t + F(X_t)) dt + G(X_t) dW_t, \quad X_0 = x \in \mathcal{H}, \qquad (12.1)$$

where:

* $\mathcal{H}$ is a separable Hilbert space,

* $A$ is the generator of a strongly continuous semigroup $(S_t)_{t \ge 0}$,

* $W_t$ is a cylindrical Wiener process on $\mathcal{H}$,

* $F$ and $G$ are nonlinear drift and diffusion maps.

The challenge lies in defining, constructing, and analyzing the process $(X_t)$ as an $\mathcal{H}$-valued stochastic process with well-defined regularity and Markov properties.

## 12.2 Stochastic Integration in Hilbert Spaces

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a complete filtered probability space satisfying the usual conditions of right-continuity and completeness. Let $\mathcal{H}$ be a separable Hilbert space with inner product $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ and norm $\| \cdot \|_{\mathcal{H}}$.
**Cylindrical Wiener process.** A cylindrical Wiener process $(W_t)_{t \ge 0}$ on $\mathcal{H}$ is a family of linear maps

$$W_t: \mathcal{H} \to L^2(\Omega, \mathcal{F}, \mathbb{P}), \quad t \ge 0,$$

such that for every $h \in \mathcal{H}$, the real-valued process $\{W_t(h)\}_{t \ge 0}$ is a standard Brownian motion satisfying

$$\mathbb{E}[W_t(h)] = 0, \quad \mathrm{Cov}(W_s(h_1), W_t(h_2)) = (s \wedge t) \langle h_1, h_2 \rangle_{\mathcal{H}}.$$

Equivalently, if $\{e_k\}_{k \ge 1}$ is a fixed orthonormal basis of $\mathcal{H}$ and $\{\beta_k(t)\}_{k \ge 1}$ is a family of independent standard Brownian motions, one can write formally

$$W_t = \sum_{k=1}^{\infty} \beta_k(t) e_k, \quad t \ge 0, \qquad (12.2)$$

where the sum is not convergent in $\mathcal{H}$ but defines a Gaussian process in a larger space of generalized random variables.

**Hilbert–Schmidt operators.** Let $\mathcal{L}_2(\mathcal{H})$ denote the space of Hilbert–Schmidt operators on $\mathcal{H}$, i.e.

$$\mathcal{L}_2(\mathcal{H}) = \{T: \mathcal{H} \to \mathcal{H} \text{ linear and bounded } | \|T\|_{\mathcal{L}_2(\mathcal{H})}^2 = \sum_{k=1}^{\infty} \|Te_k\|_{\mathcal{H}}^2 < \infty \}.$$

This space is a separable Hilbert space endowed with the inner product

$$\langle S, T \rangle_{\mathcal{L}_2(\mathcal{H})} = \sum_{k=1}^{\infty} \langle Se_k, Te_k \rangle_{\mathcal{H}},$$

independent of the chosen orthonormal basis $\{e_k\}$.

**Predictable operator-valued processes.** A process $\Phi = (\Phi_t)_{t \in [0,T]}$ is said to be *predictable* with values in $\mathcal{L}_2(\mathcal{H})$ if

$$\Phi: \Omega \times [0, T] \to \mathcal{L}_2(\mathcal{H})$$

is measurable with respect to the predictable $\sigma$-algebra $\mathcal{P}$ on $\Omega \times [0, T]$ and satisfies

$$\mathbb{E} \int_{0}^{T} \| \Phi_t \|_{\mathcal{L}_2(\mathcal{H})}^2 dt < \infty.$$

The set of all such processes is denoted by $L_{\mathcal{P}}^2([0, T]; \mathcal{L}_2(\mathcal{H}))$.

**Definition of the stochastic integral.** For any $\Phi \in L_{\mathcal{P}}^2([0, T]; \mathcal{L}_2(\mathcal{H}))$, the stochastic integral of $\Phi$ with respect to the cylindrical Wiener process $W_t$ is defined as

$$\int_{0}^{t} \Phi_{s} dW_{s} := \sum_{k=1}^{\infty} \int_{0}^{t} \Phi_{s} e_{k} d\beta_{k}(s), \qquad (12.3)$$

where the stochastic integrals on the right-hand side are Itô integrals in the sense of real-valued stochastic calculus. The convergence of the series is in $L^2(\Omega; \mathcal{H})$, since

$$\mathbb{E} \left\| \sum_{k=1}^{n} \int_{0}^{t} \Phi_s e_k d\beta_k(s) - \sum_{k=1}^{m} \int_{0}^{t} \Phi_s e_k d\beta_k(s) \right\|_{\mathcal{H}}^{2} \to 0 \quad \text{as } n, m \to \infty.$$
Hence, $\int_0^t \Phi_s dW_s$ is well-defined as an element of $L^2(\Omega; \mathcal{H})$.

**Itô isometry.** The stochastic integral satisfies the fundamental Itô isometry:

$$ \mathbb{E} \left[ \left\| \int_0^t \Phi_s dW_s \right\|_{\mathcal{H}}^2 \right] = \mathbb{E} \left[ \int_0^t \| \Phi_s \|_{\mathcal{L}_2(\mathcal{H})}^2 ds \right]. \quad (12.4) $$

*Proof.* By the definition of the stochastic integral and the orthogonality of Brownian motions $\beta_k$,

$$ \mathbb{E} \left\| \int_0^t \Phi_s dW_s \right\|_{\mathcal{H}}^2 = \mathbb{E} \left[ \sum_{k=1}^{\infty} \left\| \int_0^t \Phi_s e_k d\beta_k(s) \right\|_{\mathcal{H}}^2 \right] = \mathbb{E} \left[ \sum_{k=1}^{\infty} \int_0^t \| \Phi_s e_k \|_{\mathcal{H}}^2 ds \right], $$

which, by Fubini's theorem, yields

$$ \mathbb{E} \left[ \int_0^t \| \Phi_s \|_{\mathcal{L}_2(\mathcal{H})}^2 ds \right]. $$

This proves the identity.

**Interpretation and applications.** The process $\int_0^t \Phi_s dW_s$ is an $\mathcal{H}$-valued square-integrable martingale with quadratic variation

$$ \llbracket M \rrbracket_t = \int_0^t \| \Phi_s \|_{\mathcal{L}_2(\mathcal{H})}^2 ds. $$

This construction provides the rigorous analytic foundation for stochastic integration in infinite-dimensional spaces, thereby allowing the formulation of stochastic evolution equations (SPDEs) of the form

$$ dX_t = AX_t dt + G dW_t, \quad X_0 = x_0 \in \mathcal{H}, \quad (12.5) $$

where $A$ is a (possibly unbounded) generator of a $C_0$-semigroup on $\mathcal{H}$ and $\Phi_t$ is an admissible Hilbert-Schmidt-valued integrand. Such equations form the analytical backbone of modern infinite-dimensional stochastic analysis, encompassing models in quantum field theory, fluid dynamics, and financial mathematics.

## 12.3 Mild and Weak Solutions

Consider the abstract stochastic evolution equation on a separable Hilbert space $\mathcal{H}$,

$$ dX_t = AX_t dt + G dW_t, \quad X_0 = x \in \mathcal{H}, \quad (12.6) $$

where

* $A : \mathcal{D}(A) \subset \mathcal{H} \to \mathcal{H}$ is the infinitesimal generator of a strongly continuous semigroup $(S_t)_{t \ge 0}$ on $\mathcal{H}$,

* $G \in \mathcal{L}_2(\mathcal{H})$ is a bounded Hilbert-Schmidt operator representing the diffusion coefficient, and

* $(W_t)_{t \ge 0}$ is a cylindrical Wiener process on $\mathcal{H}$.
### 12.3.1 Mild solution

Formally integrating the stochastic differential equation from 0 to $t$ and applying the variation-of-constants (Duhamel) principle, we obtain

$$X_t = S_t x + \int_0^t S_{t-s} A X_s ds + \int_0^t S_{t-s} G dW_s.$$

However, when $A$ is unbounded, the deterministic integral may not be well-defined in the classical sense. Hence, one defines the mild solution directly as

$$X_t = S_t x + \int_0^t S_{t-s} G \, dW_s, \quad t \ge 0. \qquad (12.7)$$

The stochastic convolution term

$$W_A(t) := \int_0^t S_{t-s} G \, dW_s$$

is an $\mathcal{H}$-valued Gaussian process with mean zero and covariance operator

$$Q_t = \int_0^t S_s G G^* S_s^* ds.$$

By the Itô isometry, this process satisfies

$$\mathbb{E} \| W_A(t) \|_{\mathcal{H}}^2 = \int_0^t \| S_{t-s} G \|_{L_2(\mathcal{H})}^2 ds.$$

If the above integral is finite for all $t > 0$, then $W_A(t)$ is well-defined, and thus the mild solution $X_t$ exists uniquely. Moreover, under these conditions, $(X_t)_{t \ge 0}$ is an $\mathcal{H}$-valued adapted process satisfying

$$\mathbb{E} \| X_t \|_{\mathcal{H}}^2 < \infty, \quad \forall t \ge 0.$$

### 12.3.2 Weak solution

Let $A^*$ denote the adjoint of $A$ with domain $\mathcal{D}(A^*) \subset \mathcal{H}$. A progressively measurable process $(X_t)_{t \ge 0}$ with values in $\mathcal{H}$ is called a weak solution to the above stochastic evolution equation if, for every test function $\phi \in \mathcal{D}(A^*)$, the following holds $\mathbb{P}$-almost surely for all $t \ge 0$:

$$\langle X_t, \phi \rangle_{\mathcal{H}} = \langle x, \phi \rangle_{\mathcal{H}} + \int_0^t \langle X_s, A^* \phi \rangle_{\mathcal{H}} ds + \int_0^t \langle G dW_s, \phi \rangle_{\mathcal{H}}. \qquad (12.8)$$

The last term is well-defined as an Itô stochastic integral in $\mathbb{R}$ since

$$\langle G \, dW_s, \phi \rangle_{\mathcal{H}} = \sum_{k=1}^{\infty} \langle G e_k, \phi \rangle_{\mathcal{H}} d\beta_k(s) \qquad (12.9)$$

for the orthonormal basis $\{e_k\}$ of $\mathcal{H}$ used in the cylindrical decomposition of $W_t$.

### 12.3.3 Equivalence between mild and weak solutions

Under standard assumptions—namely:
• the operator A generates a $C_0$-semigroup ($S_t$) on $\mathcal{H}$,

• the operator G is Hilbert–Schmidt, and

• $\int_0^t \|S_{t-s}G\|_{L_2(\mathcal{H})}^2 ds < \infty$ for all $t > 0$,

the mild solution $X_t = S_t x + \int_0^t S_{t-s} G \, dW_s$ is also a weak solution in the above sense. To see this, note that for $\phi \in \mathcal{D}(A^*)$,

$$ \langle X_t, \phi \rangle_{\mathcal{H}} = \langle S_t x, \phi \rangle_{\mathcal{H}} + \int_0^t \langle S_{t-s} G \, dW_s, \phi \rangle_{\mathcal{H}} = \langle x, S_t^* \phi \rangle_{\mathcal{H}} + \int_0^t \langle G \, dW_s, S_{t-s}^* \phi \rangle_{\mathcal{H}}, $$

and using that $\frac{d}{dt}S_t^*\phi = S_t^*A^*\phi$, integration by parts yields the weak formulation.

Conversely, if ($X_t$) is a weak solution satisfying appropriate integrability and measurability conditions, then the mild representation holds via the variation-of-constants formula. Hence, under these assumptions, the notions of mild and weak solutions are equivalent.

This equivalence is fundamental in the study of stochastic evolution equations, as the mild formulation provides an explicit representation through stochastic convolution, while the weak formulation offers a variational characterization useful for duality and regularity analysis.

## 12.4 The Linear Stochastic Heat Equation

We now examine one of the most fundamental examples of an infinite-dimensional stochastic evolution equation: the *stochastic heat equation* driven by additive space–time white noise. This model serves as a canonical example illustrating the abstract theory of stochastic integration and mild solutions in Hilbert spaces.

### Formulation of the problem

Let $\mathcal{H} = L^2(0,1)$ be the Hilbert space of square-integrable real-valued functions on the unit interval $(0,1)$, equipped with the standard inner product

$$ \langle f,g \rangle_{L^2(0,1)} = \int_0^1 f(x)g(x) \, dx. \qquad (12.10) $$

We consider the stochastic partial differential equation (SPDE)

$$ \partial_t u(t,x) = \Delta u(t,x) + \dot{W}(t,x), \quad (t,x) \in (0, \infty) \times (0,1), \qquad (12.11) $$

subject to the Dirichlet boundary conditions

$$ u(t,0) = u(t,1) = 0, \quad t \ge 0, \qquad (12.12) $$

and an initial condition

$$ u(0,x) = u_0(x), \quad x \in (0,1). \qquad (12.13) $$

Here $\Delta = \frac{\partial^2}{\partial x^2}$ denotes the Laplace operator and $\dot{W}(t,x)$ represents a Gaussian noise that is white in both space and time, formally satisfying

$$ \mathbb{E}[\dot{W}(t,x)\dot{W}(s,y)] = \delta(t-s)\delta(x-y). \qquad (12.14) $$
Abstract reformulation

We now recast the SPDE into an abstract stochastic evolution equation in the Hilbert space $\mathcal{H}$.
Define the (unbounded) operator $A : \mathcal{D}(A) \subset \mathcal{H} \to \mathcal{H}$ by

$$
Au = \Delta u, \quad \mathcal{D}(A) = H^2(0,1) \cap H_0^1(0,1), \qquad (12.15)
$$

where $H^2(0, 1)$ and $H_0^1(0, 1)$ are Sobolev spaces incorporating the Dirichlet boundary conditions.
It is well known that $A$ generates a strongly continuous analytic semigroup $(S_t)_{t \ge 0}$ on $\mathcal{H}$ given
by

$$
S_t = e^{tA}. \tag{12.16}
$$

Let $(W_t)_{t \ge 0}$ be a cylindrical Wiener process on $\mathcal{H}$, i.e.

$$
W_t = \sum_{k=1}^{\infty} \beta_k(t) \phi_k, \qquad (12.17)
$$

where {$\beta_k(t)$}$_{k \ge 1}$ are independent real-valued standard Brownian motions, and {$\phi_k$}$_{k \ge 1}$ is a
complete orthonormal system in $\mathcal{H}$. Then the SPDE can be expressed compactly as

$$
du_t = Au_t dt + dW_t, \quad u_0 = u_0 \in \mathcal{H}. \tag{12.18}
$$

Mild solution

Formally integrating both sides and applying the semigroup property, the mild solution is defined
by

$$
u_t = S_t u_0 + \int_0^t S_{t-s} dW_s. \tag{12.19}
$$

The stochastic integral is well-defined in $\mathcal{H}$ since $S_{t-s}$ is a Hilbert-Schmidt operator from $\mathcal{H}$ into itself for $t > s$. Using the Itô isometry for $\mathcal{H}$-valued integrals, we obtain

$$
\mathbb{E} \| u_t \|_{\mathcal{H}}^2 = \int_0^t \| S_{t-s} \|_{L_2(\mathcal{H})}^2 ds < \infty. \quad (12.20)
$$

Hence, the mild solution $u_t$ is a well-defined $\mathcal{H}$-valued random variable for every $t \ge 0$.

Spectral decomposition

The eigenfunctions of the Dirichlet Laplacian on (0, 1) are

$$
\phi_k(x) = \sqrt{2} \sin(k\pi x), \quad k \in \mathbb{N}, \qquad (12.21)
$$

which form an orthonormal basis of $L^2(0, 1)$, with corresponding eigenvalues

$$
\lambda_k = -k^2 \pi^2. \tag{12.22}
$$

Thus, the semigroup $(S_t)$ acts diagonally in this basis:

$$
S_t \phi_k = e^{\lambda_k t} \phi_k = e^{-k^2 \pi^2 t} \phi_k. \quad (12.23)
$$
Expanding $u_t$ in this basis,

$$u_t(x) = \sum_{k=1}^{\infty} u_k(t) \phi_k(x), \qquad (12.24)$$

the SPDE decouples into an infinite system of one-dimensional stochastic differential equations:

$$du_k(t) = -k^2\pi^2 u_k(t) dt + d\beta_k(t), \quad u_k(0) = 0. \qquad (12.25)$$

Each $u_k(t)$ satisfies

$$u_k(t) = \int_0^t e^{-k^2 \pi^2 (t-s)} d\beta_k(s). \qquad (12.26)$$

Therefore, the mild solution admits the explicit representation

$$u_t(x) = \sum_{k=1}^{\infty} \left( \int_0^t e^{-k^2 \pi^2 (t-s)} d\beta_k(s) \right) \phi_k(x). \qquad (12.27)$$

## Covariance structure and Gaussianity

Since each $u_k(t)$ is a Gaussian random variable with mean zero and variance

$$\mathbb{E}[u_k(t)^2] = \int_0^t e^{-2k^2 \pi^2 (t-s)} ds = \frac{1 - e^{-2k^2 \pi^2 t}}{2k^2 \pi^2}, \qquad (12.28)$$

the process $u_t$ is a centered Gaussian random element in $\mathcal{H} = L^2(0, 1)$ with covariance operator $Q_t$ given by

$$Q_t = \int_0^t S_{t-s} S_{t-s}^* ds. \qquad (12.29)$$

In the spectral basis $\{\phi_k\}$, this covariance operator is diagonal:

$$Q_t \phi_k = \frac{1 - e^{-2k^2 \pi^2 t}}{2k^2 \pi^2} \phi_k. \qquad (12.30)$$

Hence,

$$\mathbb{E}\|u_t\|_{\mathcal{H}}^2 = \sum_{k=1}^{\infty} \frac{1 - e^{-2k^2 \pi^2 t}}{2k^2 \pi^2} < \infty. \qquad (12.31)$$

## Regularity properties

The process $(u_t)_{t \ge 0}$ is mean-square continuous in $\mathcal{H}$ and has trajectories that are almost surely continuous in time with values in $L^2(0, 1)$. However, due to the irregular nature of the space-time white noise, spatial regularity is limited: $u_t(\cdot)$ belongs almost surely to Sobolev spaces $H^\alpha(0, 1)$ only for $\alpha < \frac{1}{2}$.

Thus, the linear stochastic heat equation provides a canonical illustration of the interplay between semigroup theory, stochastic integration in Hilbert spaces, and the spectral analysis of SPDEs.
## 12.5 Nonlinear SPDEs and Monotone Operators

We now consider nonlinear stochastic partial differential equations (SPDEs) of the form

$$dX_t = (AX_t + F(X_t)) dt + G(X_t) dW_t, \quad X_0 = x \in \mathcal{H}, \qquad (12.32)$$

where A is typically a (possibly unbounded) linear operator generating a $C_0$-semigroup on a separable Hilbert space $\mathcal{H}$, and F and G are nonlinear mappings satisfying appropriate monotonicity, coercivity, and growth conditions. The driving noise $(W_t)_{t \ge 0}$ is a cylindrical Wiener process on $\mathcal{H}$ defined on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$.

### Variational setting and Gelfand triple

To rigorously treat nonlinear SPDEs, one employs the *variational* or *monotone operator* framework. Let $V$ be a reflexive Banach space continuously and densely embedded in a separable Hilbert space $\mathcal{H}$, which in turn is continuously embedded in the dual space $V^*$. This configuration forms the *Gelfand triple*

$$V \subset \mathcal{H} \subset V^*, \qquad (12.33)$$

where the embedding $V \hookrightarrow \mathcal{H}$ is continuous and dense, and we identify $\mathcal{H}$ with its dual via the Riesz representation theorem. Hence, for $v \in V$ and $h \in \mathcal{H}$,

$$\langle v, h \rangle_{V, V^*} = (v, h)_{\mathcal{H}}. \qquad (12.34)$$

### Monotonicity and coercivity conditions

The key to establishing existence and uniqueness for (12.32) lies in verifying structural properties of the drift term $A + F$. Specifically, we assume:

1. **Hemicontinuity:** For all $u, v, w \in V$, the map

$$\lambda \mapsto \langle A(u + \lambda v) + F(u + \lambda v), w \rangle_{V, V^*} \qquad (12.35)$$

is continuous on $\mathbb{R}$.

2. **Monotonicity:** There exists a constant $C \in \mathbb{R}$ such that for all $x, y \in V$,

$$\langle F(x) - F(y), x - y \rangle_{V, V^*} \le C \|x - y\|_{\mathcal{H}}^2. \qquad (12.36)$$

3. **Coercivity:** There exist constants $\alpha > 0$ and $\beta \ge 0$ such that for all $x \in V$,

$$\langle Ax + F(x), x \rangle_{V, V^*} \le -\alpha \|x\|_V^2 + \beta. \qquad (12.37)$$

4. **Growth condition:** There exists $K > 0$ such that for all $x \in V$,

$$\|Ax + F(x)\|_{V^*} \le K(1 + \|x\|_V). \qquad (12.38)$$
For the diffusion coefficient $G: V \to \mathcal{L}_2(\mathcal{H})$ (Hilbert-Schmidt operators on $\mathcal{H}$), we require:

$$ \|G(x) - G(y)\|_{\mathcal{L}_2(\mathcal{H})}^2 \le L_G \|x - y\|_{\mathcal{H}}^2, \qquad (12.39) $$

$$ \|G(x)\|_{\mathcal{L}_2(\mathcal{H})}^2 \le C_G(1 + \|x\|_{\mathcal{H}}^2), \qquad (12.40) $$

for some constants $L_G, C_G > 0$.

## Definition of variational solution

A predictable process $(X_t)_{t \ge 0}$ taking values in $V$ is called a *variational (or weak) solution* to (12.32) if, for every test function $\phi \in V$, it satisfies almost surely:

$$ \langle X_t, \phi \rangle_{V,V^*} = \langle x, \phi \rangle_{V,V^*} + \int_0^t \langle AX_s + F(X_s), \phi \rangle_{V,V^*} ds + \int_0^t \langle G(X_s) dW_s, \phi \rangle_{\mathcal{H}}. \qquad (12.41) $$

Additionally, $X_t$ must satisfy the integrability conditions:

$$ X \in L^2(\Omega; C([0, T]; \mathcal{H})) \cap L^2(\Omega; L^2(0, T; V)). \qquad (12.42) $$

## Existence and uniqueness theorem

Under the above hypotheses (12.36)-(12.40), there exists a unique variational solution $X_t$ to (12.32) satisfying

$$ \mathbb{E} \left[ \sup_{t \in [0, T]} \|X_t\|_{\mathcal{H}}^2 + \int_0^T \|X_t\|_{V}^2 dt \right] < \infty. \qquad (12.43) $$

The proof is based on a Galerkin approximation combined with monotonicity methods (Minty-Browder technique), Itô's formula, and weak compactness arguments in the variational framework.

## Energy estimate

Applying Itô's formula to $\|X_t\|_{\mathcal{H}}^2$, we obtain:

$$ d\|X_t\|_{\mathcal{H}}^2 = 2\langle AX_t + F(X_t), X_t \rangle_{V,V^*} dt + \|G(X_t)\|_{\mathcal{L}_2(\mathcal{H})}^2 dt + 2\langle X_t, G(X_t) dW_t \rangle_{\mathcal{H}}. \qquad (12.44) $$

Taking expectations and using (12.37) and (12.40), we obtain the a priori estimate

$$ \mathbb{E}\|X_t\|_{\mathcal{H}}^2 + 2\alpha\mathbb{E}\int_0^t \|X_s\|_V^2 ds \le \|x\|_{\mathcal{H}}^2 + C\int_0^t (1 + \mathbb{E}\|X_s\|_{\mathcal{H}}^2) ds. \qquad (12.45) $$

By Grönwall's inequality, this yields

$$ \sup_{t \in [0, T]} \mathbb{E} \|X_t\|_{\mathcal{H}}^2 < \infty. \qquad (12.46) $$

## Examples

This general framework includes several important nonlinear SPDEs as special cases:

* **Stochastic porous medium equation:**

$$ du_t = \Delta(u_t^m) dt + \sigma(u_t) dW_t, \quad m > 1. \qquad (12.47) $$
* Stochastic reaction-diffusion equation:

$$du_t = (\Delta u_t + f(u_t)) dt + g(u_t) dW_t. \qquad (12.48)$$

* Stochastic Navier-Stokes equations (in 2D):

$$du_t = [\nu \Delta u_t - (u_t \cdot \nabla) u_t - \nabla p_t] dt + B(u_t) dW_t, \quad \nabla \cdot u_t = 0. \qquad (12.49)$$

Hence, the variational framework for monotone SPDEs provides a unified and rigorous setting encompassing a broad class of nonlinear infinite-dimensional stochastic systems, ensuring existence, uniqueness, and energy stability of their solutions.

## 12.6 Invariant Measures and Ergodicity

Let $(X_t)_{t \ge 0}$ be a Markov process on a separable Hilbert space $\mathcal{H}$ with transition semigroup $(P_t)_{t \ge 0}$ acting on bounded measurable functions $f: \mathcal{H} \to \mathbb{R}$ via

$$(P_t f)(x) = \mathbb{E}^x[f(X_t)].$$

A probability measure $\mu$ on $\mathcal{H}$ is said to be an *invariant measure* for $(P_t)$ if, for every bounded measurable $f: \mathcal{H} \to \mathbb{R}$,

$$\int_{\mathcal{H}} (P_t f)(x) \mu(dx) = \int_{\mathcal{H}} f(x) \mu(dx), \quad \forall t \ge 0. \qquad (12.50)$$

Equivalently, if $X_0 \sim \mu$, then $X_t \sim \mu$ for all $t \ge 0$, i.e. $\mu$ is stationary for the process.

### Gaussian invariant measure for linear systems

Consider the linear stochastic evolution equation on $\mathcal{H}$:

$$dX_t = AX_t dt + G dW_t, \quad X_0 = x \in \mathcal{H}, \qquad (12.51)$$

where:

* $A: \mathcal{D}(A) \subset \mathcal{H} \to \mathcal{H}$ is the infinitesimal generator of a strongly continuous semigroup $(S_t)_{t \ge 0}$ on $\mathcal{H}$,

* $G \in \mathcal{L}_2(\mathcal{H})$ is a bounded Hilbert-Schmidt operator (diffusion coefficient),

* $(W_t)_{t \ge 0}$ is a cylindrical Wiener process on $\mathcal{H}$.

The mild solution of (12.51) is

$$X_t = S_t x + \int_0^t S_{t-s} G dW_s. \qquad (12.52)$$

This process $(X_t)$ is known as the *Ornstein-Uhlenbeck process* in infinite dimensions.
## Existence of invariant measure

Assume that A is dissipative, meaning

$$ \langle Ax, x \rangle_{\mathcal{H}} \leq -\omega \|x\|_{\mathcal{H}}^2, \quad \forall x \in \mathcal{D}(A), \qquad (12.53) $$

for some $\omega > 0$. This implies that $\|S_t\|_{\mathcal{L}(\mathcal{H})} \leq e^{-\omega t}$, hence $(S_t)$ is exponentially stable.

Define the covariance operator

$$ Q_{\infty} := \int_{0}^{\infty} S_{t}GG^{*}S_{t}^{*} dt. \qquad (12.54) $$

Because $S_t$ decays exponentially, the integral converges in the trace-class operator sense. The operator $Q_\infty$ is self-adjoint, positive semidefinite, and satisfies the Lyapunov equation

$$ AQ_{\infty} + Q_{\infty}A^{*} + GG^{*} = 0. \qquad (12.55) $$

To verify this, note that differentiating the finite-time covariance

$$ Q_t = \int_0^t S_s GG^* S_s^* ds $$

gives

$$ \frac{dQ_t}{dt} = S_t GG^* S_t^* = AQ_t + Q_t A^* + GG^*, $$

and taking the limit as $t \to \infty$ (using the stability of $S_t$) yields (12.55).

## Characterization of the invariant measure

Define the Gaussian measure $\mu_\infty := \mathcal{N}(0, Q_\infty)$ on $\mathcal{H}$ with mean zero and covariance operator $Q_\infty$. Then $\mu_\infty$ satisfies (12.50), i.e.

$$ \int_{\mathcal{H}} (P_t f)(x) \mu_{\infty}(dx) = \int_{\mathcal{H}} f(x) \mu_{\infty}(dx), \quad \forall t \geq 0. $$

Indeed, since $X_t$ is Gaussian with mean $S_t x$ and covariance $Q_t$, if $X_0 \sim \mu_\infty$ then

$$ X_t \sim \mathcal{N}(S_t x, Q_t + S_t Q_\infty S_t^*), $$

and using the identity $Q_t + S_t Q_\infty S_t^* = Q_\infty$ (which follows from (12.54)) we deduce stationarity of $\mu_\infty$.

## Uniqueness and ergodicity

If the pair $(A, G)$ is controllable in the sense that

$$ \overline{\text{Range}(Q_\infty^{1/2})} = \mathcal{H}, $$

then $\mu_\infty$ has full support on $\mathcal{H}$ and is the unique invariant probability measure of $(X_t)$.
Furthermore, ($X_t$) is ergodic with respect to $\mu_\infty$, i.e. for every $f \in L^1(\mathcal{H}, \mu_\infty)$,

$$ \lim_{T \to \infty} \frac{1}{T} \int_0^T f(X_t) dt = \int_{\mathcal{H}} f(x) \mu_\infty(dx) \quad \text{a.s.} \qquad (12.56) $$

This property follows from the exponential stability of $S_t$ and mixing of the Gaussian process.

## Summary: the Ornstein–Uhlenbeck semigroup

The semigroup ($P_t$) associated with (12.51) acts on bounded continuous functions via

$$ (P_t f)(x) = \int_{\mathcal{H}} f(S_t x + y) \mathcal{N}(0, Q_t)(dy), $$

and is strongly Feller and irreducible when $G$ is non-degenerate. It satisfies the generator identity

$$ \mathcal{L}f(x) = \langle Ax, Df(x) \rangle_{\mathcal{H}} + \frac{1}{2} \mathrm{Tr} (GG^*D^2f(x)), $$

and admits $\mu_\infty = \mathcal{N}(0, Q_\infty)$ as its unique invariant measure.

Thus, under dissipativity of $A$ and non-degeneracy of $G$, the infinite-dimensional Ornstein–Uhlenbeck process is Gaussian, stationary, ergodic, and fully characterized by the Lyapunov equation (12.55).

## 12.7 Malliavin Calculus and Regularity of Laws

Malliavin calculus furnishes a differential calculus on Wiener space which permits one to study regularity of probability laws of functionals of Gaussian processes, and — by extension — of solutions to stochastic (partial) differential equations. In what follows we work on a probability space carrying a cylindrical (or genuine) Wiener process $W$ taking values in a separable Hilbert space $\mathcal{U}$. Denote by $\mathcal{H}$ the Cameron–Martin space associated with $W$ (for an $\mathcal{U}$-cylindrical Wiener process on $[0, T]$ one typically has $\mathcal{H} = L^2([0, T]; \mathcal{U})$). We adopt the usual Wiener–Itô isometry identifications and use $\langle \cdot, \cdot \rangle_{\mathcal{H}}$ for the inner product on $\mathcal{H}$.

Let $F$ be a smooth cylindrical functional of the form

$$ F = f(W(h_1), \dots, W(h_n)), \qquad (12.57) $$

where $h_i \in \mathcal{H}$, $W(h) = \int_0^T \langle h(s), dW_s \rangle_{\mathcal{H}}$, and $f \in C_b^\infty(\mathbb{R}^n)$. The Malliavin derivative $D$ is defined on such $F$ by the $\mathcal{H}$-valued random variable

$$ (DF)(\omega) = \sum_{i=1}^{n} \partial_i f(W(h_1), \dots, W(h_n)) h_i \in \mathcal{H}. \qquad (12.58) $$

Iterating, one obtains higher-order derivatives $D^k F \in L^2(\Omega; \mathcal{H}^{\otimes k})$. The derivative operator $D$ is closable from $L^p(\Omega)$ into $L^p(\Omega; \mathcal{H})$, and one defines the Sobolev–Malliavin spaces

$$ \mathbb{D}^{k,p} := \overline{\{ \text{smooth cylindrical } F \}}^{\|\cdot\|_{k,p}}, \qquad \|F\|_{k,p}^p := \mathbb{E}[|F|^p] + \sum_{j=1}^k \mathbb{E}[\|D^j F\|_{\mathcal{H}^{\otimes j}}^p]. \qquad (12.59) $$
The adjoint of $D$ (in the sense of the $L^2$-pairing) is the divergence (or Skorohod) operator $\delta$. For an adapted (or more generally predictable) $\mathcal{H}$-valued process $u$ satisfying suitable integrability, $\delta(u)$ coincides with the Itô integral $\int_0^T \langle u_s, dW_s \rangle_u$; for nonadapted $u$ the divergence extends this integral in the Skorohod sense. The fundamental duality formula reads, for appropriate $F \in \mathbb{D}^{1,2}$ and $u \in \text{Dom}(\delta)$,

$$ \mathbb{E}[\langle DF, u \rangle_{\mathcal{H}}] = \mathbb{E}[F \delta(u)]. \qquad (12.60) $$

A principal application is the study of existence and smoothness of densities of random vectors. Let $F = (F^1, \dots, F^m)$ be an $\mathbb{R}^m$-valued random variable with components in $\mathbb{D}^{1,2}$. Its Malliavin covariance matrix $\sigma_F$ is the $m \times m$ symmetric, nonnegative-definite random matrix

$$ \sigma_F := (\langle DF^i, DF^j \rangle_{\mathcal{H}})_{1 \le i,j \le m}. \qquad (12.61) $$

A cornerstone result (Bouleau-Hirsch) states that if $F \in \mathbb{D}^{1,2}$ and $\sigma_F$ is almost surely invertible with inverse having moments of all orders (or at least $\mathbb{E}[\det(\sigma_F)^{-p}] < \infty$ for some $p > 1$), then the law of $F$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}^m$. Moreover, higher-order Malliavin differentiability together with bounds on $\sigma_F^{-1}$ yield $C^\infty$-smoothness of the density. Precisely, if $F \in \bigcap_{k \ge 1} \mathbb{D}^{k,p}$ for all $p$ and $\det \sigma_F^{-1} \in \bigcap_{p \ge 1} L^p(\Omega)$, then the density of $F$ belongs to $C^\infty(\mathbb{R}^m)$.

In infinite-dimensional settings one rarely seeks a density with respect to an infinite-dimensional “Lebesgue” measure; instead one studies finite-dimensional projections or densities with respect to Gaussian/reference measures. Thus, for an $\mathcal{H}$-valued random variable $X$ (e.g. the solution of an SPDE), one typically examines the law of $\ell(X)$ for finite-rank linear functionals $\ell: \mathcal{H} \to \mathbb{R}^m$. If $X$ is Malliavin differentiable and the projected covariance

$$ \sigma_{\ell(X)} = (\langle D\langle X, \ell_i \rangle, D\langle X, \ell_j \rangle\rangle_{\mathcal{H}})_{i,j} \qquad (12.62) $$

is nondegenerate in the Bouleau-Hirsch sense, then $\ell(X)$ admits a smooth density.

A central class of applications concerns solutions to (finite- or infinite-dimensional) SDEs/SPDEs. Consider the abstract stochastic evolution equation

$$ dX_t = (AX_t + F(X_t)) dt + G(X_t) dW_t, \quad X_0 = x, \qquad (12.63) $$

posed in a Hilbert (or Banach) space. Under sufficient regularity of $F$ and $G$ (Fréchet differentiability, Lipschitz-type bounds) one can prove that the solution map $\omega \mapsto X_t(\omega)$ is Malliavin differentiable, and the Malliavin derivative $D_s X_t \in \mathcal{L}(\mathcal{H}; \mathcal{H})$ (or an $\mathcal{H}$-valued kernel) satisfies the linearized (variation) equation obtained by differentiating the SPDE in direction $h \in \mathcal{H}$. For $0 \le s \le t$ one has, in mild form,

$$ D_s X_t = S_{t-s} G(X_s) + \int_s^t S_{t-r} \nabla F(X_r) D_s X_r dr + \int_s^t S_{t-r} \nabla G(X_r) D_s X_r dW_r, \qquad (12.64) $$

where $S_t$ denotes the semigroup generated by $A$, and $\nabla F, \nabla G$ are appropriate Fréchet derivatives (these integrals are to be interpreted in the Ito/Skorohod sense depending on adaptivity). When $G$ is nondegenerate and the linearized equation is sufficiently controllable, one may de-
duce nondegeneracy of the finite-dimensional Malliavin covariance matrices.

In finite dimensions the celebrated Hörmander condition yields hypoellipticity: if the Lie algebra generated by the vector fields appearing in the SDE spans $\mathbb{R}^n$ at the point of interest, then the Malliavin matrix is invertible and the density is smooth. In infinite-dimensional or SPDE contexts one often uses finite-dimensional projection versions of Hörmander's bracket condition (or its adapted infinite-dimensional analogues) to obtain smoothness of finite-dimensional marginals. Concretely, suppose for a family of linear functionals $\ell_1, \dots, \ell_m$ the Lie-bracket-generated reachable space (formed by iterated commutators of the drift and noise directions projected by $\ell$) spans $\mathbb{R}^m$; then $\ell(X_t)$ has a smooth density.

The Malliavin covariance operator associated to an $\mathcal{H}$-valued random variable $X$ is the trace-class, symmetric operator $C_X : \mathcal{H} \to \mathcal{H}$ given by

$$C_X = \mathbb{E}[(DX) \otimes (DX)] = \mathbb{E}[(DX) \langle DX, \cdot \rangle_{\mathcal{H}}], \qquad (12.65)$$

and for a finite-rank projection $P : \mathcal{H} \to \mathbb{R}^m$ one recovers $\sigma_{P(X)} = P C_X P^*$. In many SPDEs one proves that $C_X$ is positive definite on the range of the projection, or equivalently that $\langle C_X u, u \rangle_{\mathcal{H}} > 0$ for all nonzero $u$ in a finite-dimensional subspace; this suffices to obtain absolute continuity of the projected law.

Integration by parts formulae derived from Malliavin calculus underpin density formulas and quantitative bounds. If $F \in \mathbb{D}^{1,2}$ and $\varphi \in C_c^\infty(\mathbb{R}^m)$, then repeated application of the duality between $D$ and $\delta$ yields

$$\mathbb{E}[\partial_{i_1} \cdots \partial_{i_k} \varphi(F)] = \mathbb{E}[\varphi(F) H_{i_1, \dots, i_k}(F)], \qquad (12.66)$$

where the random weights $H_{i_1, \dots, i_k}(F)$ are expressed in terms of iterated divergences involving $\sigma_F^{-1}$ and Malliavin derivatives of $F$. From these identities one deduces that the density of $F$ (when it exists) admits classical Sobolev and smoothness estimates controlled by moments of $\sigma_F^{-1}$ and norms of derivatives of $F$.

Concerning SPDEs, typical rigorous results take the following form. Suppose the coefficients $F$ and $G$ are sufficiently smooth with bounded derivatives, and that the noise acts nondegenerately in a manner satisfying a suitable controllability (or bracket) condition on a finite-dimensional projection. Then for every $t > 0$ and every finite-rank bounded linear operator $P : \mathcal{H} \to \mathbb{R}^m$, the random vector $PX_t$ belongs to $\bigcap_{k,p} \mathbb{D}^{k,p}(\mathbb{R}^m)$ and has a $C^\infty$-density with respect to Lebesgue measure on $\mathbb{R}^m$. Moreover, if the Malliavin covariance of $PX_t$ admits uniform lower bounds (elipticity) one obtains Gaussian-type lower and upper bounds for the density via Norris' lemma and heat-kernel techniques adapted to the Malliavin setting.

Finally, Malliavin calculus provides tools for quantitative regularity: existence of densities in Sobolev spaces $W^{s,p}$, short-time asymptotics, gradient estimates for densities, and bounds on derivatives of transition probabilities. These results are central in establishing Hörmander-type hypoellipticity for SPDEs, smoothing properties of their transition semigroups, and precise
control on the probabilistic representation of solutions to PDEs via probabilistic methods.

## 12.8 Spectral and Functional Analytic Structure

The study of the spectral and functional analytic properties of stochastic evolution equations is centered on the infinitesimal generator of their associated Markov semigroup. Let $(X_t^x)_{t \ge 0}$ denote the mild solution to the stochastic evolution equation

$$dX_t = (AX_t + F(X_t)) dt + G dW_t, \quad X_0 = x \in \mathcal{H}, \qquad (12.67)$$

where $A$ is the generator of a $C_0$-semigroup $(S_t)_{t \ge 0}$ on a separable Hilbert space $\mathcal{H}$, $F: \mathcal{H} \to \mathcal{H}$ is a (possibly nonlinear) drift term satisfying suitable Lipschitz and growth conditions, and $G \in \mathcal{L}(\mathcal{U}, \mathcal{H})$ is a bounded linear operator defining the covariance structure of a $\mathcal{U}$-valued cylindrical Wiener process $W_t$.

For every bounded Borel measurable function $f: \mathcal{H} \to \mathbb{R}$, the associated Markov semigroup is defined by

$$(P_t f)(x) := \mathbb{E}[f(X_t^x)], \quad t \ge 0, x \in \mathcal{H}. \qquad (12.68)$$

The family $(P_t)_{t \ge 0}$ is a strongly continuous semigroup on suitable Banach spaces of functions on $\mathcal{H}$ (typically $C_b(\mathcal{H})$, $B_b(\mathcal{H})$, or $L^p(\mu)$ spaces with respect to an invariant measure $\mu$), and its infinitesimal generator $\mathcal{L}$ is defined formally by

$$\mathcal{L}f(x) := \lim_{t \downarrow 0} \frac{P_t f(x) - f(x)}{t}, \qquad (12.69)$$

with domain $\mathcal{D}(\mathcal{L})$ consisting of all functions $f$ for which the limit exists (in the appropriate topology).

Assuming $f$ is twice continuously Fréchet differentiable with bounded derivatives, one can derive the formal expression of $\mathcal{L}$ via Itô's formula. Let $Df(x) \in \mathcal{L}(\mathcal{H}; \mathbb{R}) \simeq \mathcal{H}$ denote the first Fréchet derivative (gradient) and $D^2f(x) \in \mathcal{L}(\mathcal{H}, \mathcal{H})$ the second derivative (Hessian). Applying Itô's formula in the Hilbert space setting yields

$$df(X_t^x) = \langle Df(X_t^x), AX_t^x + F(X_t^x) \rangle_{\mathcal{H}} dt + \frac{1}{2} \mathrm{Tr}(GG^*D^2f(X_t^x)) dt + \langle Df(X_t^x), G dW_t \rangle_{\mathcal{H}}. \qquad (12.70)$$

Taking expectations and dividing by $t$, then letting $t \downarrow 0$, one obtains the expression for the generator:

$$\mathcal{L}f(x) = \langle Ax + F(x), Df(x) \rangle_{\mathcal{H}} + \frac{1}{2} \mathrm{Tr}(GG^*D^2f(x)). \qquad (12.71)$$

This operator is the infinite-dimensional analogue of the Kolmogorov backward operator for finite-dimensional diffusion processes.

**Domain and Core.** A natural choice for the core of $\mathcal{L}$ is the algebra $\mathcal{FC}_b^2(\mathcal{H})$ of smooth
cylindrical functions of the form

$$f(x) = \psi(\langle x, h_1 \rangle, \dots, \langle x, h_n \rangle), \qquad (12.72)$$

where $\psi \in C_b^2(\mathbb{R}^n)$ and $h_1, \dots, h_n \in \mathcal{H}$. For such $f$, the derivatives are given by

$$Df(x) = \sum_{i=1}^{n} \partial_i \psi(\langle x, h_1 \rangle, \dots, \langle x, h_n \rangle) h_i, \qquad (12.73)$$

$$D^2 f(x) = \sum_{i,j=1}^{n} \partial_{ij}^2 \psi(\langle x, h_1 \rangle, \dots, \langle x, h_n \rangle) h_i \otimes h_j, \qquad (12.74)$$

and thus $\mathcal{L}$ acts as a second-order differential operator on a finite-dimensional projection of $\mathcal{H}$. This definition extends to more general function spaces via closure.

**Functional Analytic Characterization.** The operator $\mathcal{L}$ is typically dissipative in $L^2(\mu)$ (for invariant measure $\mu$) if

$$\int_{\mathcal{H}} (\mathcal{L} f) f d\mu \le 0, \quad \forall f \in \mathcal{D}(\mathcal{L}), \qquad (12.75)$$

and maximally dissipative when the range condition $(\lambda I - \mathcal{L})(\mathcal{D}(\mathcal{L})) = L^2(\mu)$ holds for some $\lambda > 0$. In such case, $\mathcal{L}$ generates a contraction semigroup on $L^2(\mu)$ by the Hille–Yosida theorem.

When $A$ is self-adjoint, negative definite, and $G$ non-degenerate, the Ornstein–Uhlenbeck operator

$$\mathcal{L}_0 f(x) = \langle Ax, Df(x) \rangle + \frac{1}{2} \mathrm{Tr}(GG^* D^2 f(x)) \qquad (12.76)$$

is itself self-adjoint in $L^2(\mu)$, where $\mu$ is the invariant Gaussian measure with covariance operator $Q_\infty = \int_0^\infty S_t GG^* S_t^* dt$. In such Gaussian setting, $\mathcal{L}_0$ is related to the Gross–Ornstein–Uhlenbeck operator, generating a strongly continuous, analytic, and hypercontractive semigroup.

For more general nonlinear drifts $F$, $\mathcal{L}$ is not self-adjoint but typically sectorial in $L^2(\mu)$: there exists $\theta < \pi/2$ such that

$$\| (zI - \mathcal{L})^{-1} \|_{\mathcal{L}(L^2(\mu))} \le \frac{C}{|z|}, \quad z \in \mathbb{C} \setminus \Sigma_\theta, \qquad (12.77)$$

where $\Sigma_\theta$ is an open sector centered on the positive real axis. Sectoriality implies that $\mathcal{L}$ generates an analytic semigroup, enabling fractional powers and functional calculus.

**Spectral Structure.** The spectrum $\sigma(\mathcal{L})$ determines the long-time behavior of $(P_t)$. For instance:

* If $\mathcal{L}$ has purely discrete spectrum $\{\lambda_k\}$ with $0 = \lambda_0 > \mathrm{Re}\,\lambda_1 \ge \mathrm{Re}\,\lambda_2 \ge \dots$, then $(P_t)$ admits an eigenfunction expansion:

$$P_t f = \sum_{k=0}^{\infty} e^{\lambda_k t} \langle f, \phi_k \rangle_{\mu} \phi_k.$$
* The spectral gap $\gamma = -\sup\{\text{Re } \lambda_k : \lambda_k \neq 0\}$ quantifies the exponential rate of convergence to equilibrium.

* If $\mathcal{L}$ is self-adjoint in $L^2(\mu)$, then $\sigma(\mathcal{L}) \subset (-\infty, 0]$, and $\mathcal{L}$ admits a spectral decomposition via a resolution of the identity:

$$ \mathcal{L} = \int_{(-\infty,0]} \lambda \, dE(\lambda), $$

$$ \text{with } P_t = \int_{(-\infty,0]} e^{\lambda t} dE(\lambda). $$

These analytic and spectral properties play a central role in the study of regularity, ergodicity, and stability of infinite-dimensional diffusions. In particular, coercivity and sectoriality of $\mathcal{L}$ underpin the construction of invariant measures, while its spectral gap governs mixing rates and functional inequalities (such as Poincaré and logarithmic Sobolev inequalities) in infinite-dimensional probability spaces.

## 12.9 Connection to Dirichlet Forms

In the study of infinite-dimensional stochastic processes, particularly those arising from stochastic partial differential equations (SPDEs), Dirichlet forms play a central role in establishing the link between probabilistic and analytic structures. Let $\mu$ be a probability measure on a separable Hilbert space $\mathcal{H}$ which is invariant under a Markov semigroup $(P_t)_{t \ge 0}$ associated with an SPDE. We denote by $L^2(\mathcal{H}, \mu)$ the corresponding Hilbert space of square-integrable functions.

### Definition of the Dirichlet Form

For smooth cylindrical functions $f, g: \mathcal{H} \to \mathbb{R}$, the associated Dirichlet form is defined as

$$ \mathcal{E}(f,g) = \int_{\mathcal{H}} \langle Df(x), QDg(x) \rangle_{\mathcal{H}} \mu(dx), \qquad (12.78) $$

where $D$ denotes the Fréchet derivative, and $Q$ is a positive, self-adjoint, trace-class operator acting as the covariance operator of the Gaussian measure $\mu$. The quadratic form

$$ \mathcal{E}(f,f) = \int_{\mathcal{H}} \|Q^{1/2} Df(x)\|_{\mathcal{H}}^2 \mu(dx) \qquad (12.79) $$

encodes the “energy” associated with the function $f$ and defines a nonnegative symmetric bilinear form on a dense subspace of $L^2(\mathcal{H}, \mu)$.

### Closability and Associated Operator

The form $(\mathcal{E}, \mathcal{F})$, where $\mathcal{F}$ is the closure of smooth cylindrical functions under the norm

$$ \|f\|_{\mathcal{E}}^2 = \|f\|_{L^2(\mu)}^2 + \mathcal{E}(f,f), \qquad (12.80) $$

is closable in $L^2(\mathcal{H}, \mu)$, and its closure defines a symmetric, strongly local, regular Dirichlet form. By the general theory of Dirichlet forms (see Fukushima-Oshima-Takeda), there exists
a unique non-positive self-adjoint operator $(\mathcal{L}, D(\mathcal{L}))$ on $L^2(\mathcal{H}, \mu)$ such that

$$ \mathcal{E}(f,g) = - \int_{\mathcal{H}} (\mathcal{L}f)(x) g(x) \mu(dx), \quad f \in D(\mathcal{L}), g \in \mathcal{F}. \qquad (12.81) $$

This operator $\mathcal{L}$ coincides with the infinitesimal generator of the Markov semigroup $(P_t)$ associated with the SPDE.

### Example: Ornstein–Uhlenbeck Process

For the Ornstein–Uhlenbeck process on $\mathcal{H}$ given by

$$ dX_t = AX_t dt + G dW_t, \qquad (12.82) $$

with invariant Gaussian measure $\mu = \mathcal{N}(0, Q_\infty)$ satisfying

$$ AQ_\infty + Q_\infty A^* + GG^* = 0, \qquad (12.83) $$

the corresponding Dirichlet form takes the explicit form

$$ \mathcal{E}(f,g) = \int_{\mathcal{H}} \langle Q_{\infty}^{1/2} Df(x), Q_{\infty}^{1/2} Dg(x) \rangle_{\mathcal{H}} \mu(dx). \qquad (12.84) $$

The generator $\mathcal{L}$ associated with this form is the Ornstein–Uhlenbeck operator

$$ \mathcal{L}f(x) = \frac{1}{2}\mathrm{Tr}(Q_\infty D^2 f(x)) + \langle Ax, Df(x) \rangle_\mathcal{H}, \qquad (12.85) $$

which acts as an infinite-dimensional analogue of the classical elliptic operator associated with diffusion.

### Functional Analytic and Probabilistic Connection

The Dirichlet form framework provides a bridge between:

1. **Functional Analysis:** where $\mathcal{E}$ defines a symmetric bilinear form generating a self-adjoint operator;

2. **Potential Theory:** where $\mathcal{E}$ characterizes notions of capacity, energy, and equilibrium;

3. **Stochastic Analysis:** where $\mathcal{E}$ corresponds to the quadratic variation and energy dissipation of stochastic dynamics.

In particular, the Markov semigroup $(P_t)$ associated with the SPDE satisfies the energy identity

$$ \frac{d}{dt} \|P_t f\|_{L^2(\mu)}^2 = -2 \mathcal{E}(P_t f, P_t f), \qquad (12.86) $$

which reveals that $\mathcal{E}$ governs the dissipation of energy in the evolution of expectations.

### Conclusion

Thus, under a Gaussian invariant measure $\mu$, the Dirichlet form

$$ \mathcal{E}(f,g) = \int_{\mathcal{H}} \langle Df(x), QDg(x) \rangle_{\mathcal{H}} \mu(dx) \qquad (12.87) $$
is symmetric, closable, and generates a self-adjoint operator that coincides with the infinitesimal generator of the infinite-dimensional diffusion. This unifies the analytic, geometric, and stochastic aspects of SPDEs and provides the foundation for potential theory and ergodicity in infinite-dimensional spaces.

## 12.10 Conclusion

Stochastic partial differential equations extend the realm of diffusion to infinite-dimensional configuration spaces. Their study intertwines semigroup theory, stochastic calculus, spectral analysis, and infinite-dimensional functional analysis. Through the unifying framework of Dirichlet forms and Malliavin calculus, SPDEs reveal the profound interplay between randomness, geometry, and analysis — providing the mathematical language for noise-driven dynamics in continua and fields.

**References:** Ghosh (2025) [32], Da Prato and Zabczyk (1992) [10], Röckner et al. (2014) [38].
# Stochastic Analysis on Manifolds and Geometric Diffusions

> “Geometry is the language with which probability describes motion.”

> — Anonymous

## 13.1 Brownian Motion on Riemannian Manifolds

Let $(M, g)$ be a smooth, connected, $d$-dimensional Riemannian manifold with metric tensor $g = (g_{ij})$, inverse $g^{ij}$, and Riemannian volume form $dvol_g = \sqrt{|g|} dx$ where $|g| = \det(g_{ij})$. The Laplace-Beltrami operator $\Delta_g$ acting on $f \in C^\infty(M)$ is the divergence of the gradient,

$$ \Delta_g f = \operatorname{div}_g(\nabla f) = \frac{1}{\sqrt{|g|}} \partial_i(\sqrt{|g|} g^{ij} \partial_j f), \qquad (13.1) $$

with the Einstein summation convention in place. The operator $\frac{1}{2}\Delta_g$ is a (symmetric, nonpositive) second-order elliptic differential operator which serves as the infinitesimal generator of Brownian motion on $(M, g)$.

A stochastic process $(X_t)_{t \ge 0}$ with continuous paths and values in $M$ is called a Brownian motion on $(M, g)$ (started at $x \in M$) if for every $f \in C_c^\infty(M)$ the process

$$ M_t^f = f(X_t) - f(X_0) - \frac{1}{2} \int_0^t (\Delta_g f)(X_s) ds \qquad (13.2) $$

is a real-valued martingale with respect to the natural filtration. Equivalently, the transition semigroup

$$ (P_t f)(x) := \mathbb{E}^x[f(X_t)] \qquad (13.3) $$

is the heat semigroup solving the Cauchy problem

$$ \partial_t u(t, x) = \frac{1}{2} \Delta_g u(t, x), \quad u(0, \cdot) = f, \qquad (13.4) $$

so that $P_t = e^{\frac{t}{2}\Delta_g}$ on $L^2(M, dvol_g)$ (with appropriate domain considerations).
**Local SDE representation.** In local coordinates $(x^1, \dots, x^d)$ an orthonormal frame $\{e_a(x)\}_{a=1}^d$ (for instance obtained by orthonormalizing coordinate vector fields) gives vector fields $e_a = e_a^i(x) \partial_{x^i}$ satisfying $g(e_a, e_b) = \delta_{ab}$. If $(B_t^a)_{a=1}^d$ are independent real Brownian motions, the Stratonovich stochastic differential equation

$$dX_t = e_a(X_t) \circ dB_t^a \qquad (13.5)$$

defines (up to explosion) a diffusion whose generator equals $\frac{1}{2} \sum_{a=1}^d L_{e_a}^2 = \frac{1}{2} \Delta_g$, where $L_{e_a}$ denotes Lie differentiation along $e_a$. Writing components in local coordinates, the Itô form of the same equation is

$$dX_t^i = e_a^i(X_t) dB_t^a + \frac{1}{2} g^{jk}(X_t) \Gamma_{jk}^i(X_t) dt, \qquad (13.6)$$

where $\Gamma_{jk}^i$ are the Christoffel symbols of the Levi-Civita connection. The drift term $\frac{1}{2}g^{jk}\Gamma_{jk}^i$ is the Itô correction ensuring that the generator is exactly $\frac{1}{2}\Delta_g$.

**Existence, uniqueness and stochastic development.** Existence and uniqueness (in law) of Brownian motion on a complete Riemannian manifold follow from the construction on the orthonormal frame bundle $O(M)$. Let $\pi: O(M) \to M$ be the orthonormal frame bundle and $H_a$ the horizontal vector fields corresponding to the canonical horizontal distribution determined by the Levi-Civita connection. Solving the horizontal Stratonovich SDE on $O(M)$

$$dU_t = H_a(U_t) \circ dB_t^a, \qquad (13.7)$$

with $U_0$ an orthonormal frame at $x$, and projecting $X_t = \pi(U_t)$ yields Brownian motion on $M$. This stochastic development maps Euclidean Brownian motion to manifold Brownian motion and provides a powerful geometric construction which also yields the stochastic parallel transport along paths via the frame process $U_t$.

**Non-explosion and stochastic completeness.** A Riemannian manifold $(M, g)$ is called stochastically complete if Brownian motion started at any $x \in M$ is non-explosive (i.e. has infinite lifetime a.s.). Geometric sufficient conditions for stochastic completeness include: geodesic completeness with Ricci curvature bounded below, or the Grigor'yan volume growth condition. Analytically, stochastic completeness is equivalent to preservation of mass by the heat semigroup: $P_t1 = 1$ for all $t \ge 0$.

**Heat kernel and short-time asymptotics.** Under mild hypotheses (e.g. completeness and smoothness) the heat semigroup admits a smooth positive heat kernel $p_t(x, y)$ with respect to the Riemannian volume, so that

$$(P_t f)(x) = \int_M p_t(x, y) f(y) d\mathrm{vol}_g(y). \qquad (13.8)$$

The heat kernel is jointly $C^\infty$ on $(0, \infty) \times M \times M$ and satisfies the Chapman-Kolmogorov identity. The celebrated Minakshisundaram-Pleijel (or heat kernel) expansion gives as $t \downarrow 0$

$$p_t(x, y) \sim (4\pi t)^{-d/2} e^{-\frac{\mathrm{dist}(x, y)^2}{4t}} \sum_{k=0}^{\infty} a_k(x, y) t^k, \qquad (13.9)$$
and Varadhan's short-time principle reads

$$ \lim_{t \downarrow 0} -2t \log p_t(x, y) = \mathrm{dist}(x, y)^2, \qquad (13.10) $$

uniformly on compact sets, where $\mathrm{dist}(\cdot, \cdot)$ is the Riemannian distance.

**Geometry and analytic estimates.** Curvature controls analytic and probabilistic behaviour. If the Ricci tensor satisfies $\mathrm{Ric} \ge Kg$ for some $K \in \mathbb{R}$, then Bakry-Émery gradient estimates hold:

$$ |\nabla P_t f|^2 \le e^{-2Kt} P_t(|\nabla f|^2), \quad f \in C_b^\infty(M), \qquad (13.11) $$

and corresponding functional inequalities follow (Poincaré, log-Sobolev). Coupling constructions using parallel transport along geodesics yield Wasserstein contraction estimates:

$$ W_2(\delta_x P_t, \delta_y P_t) \le e^{-Kt} \mathrm{dist}(x, y), \qquad (13.12) $$

whenever $\mathrm{Ric} \ge K$.

**Stochastic parallel transport and connection Laplacian.** Let $\|\cdot\| : T_{X_0}M \to T_{X_t}M$ denote stochastic parallel transport along $X$ defined by the horizontal frame process $U_t$. The covariant derivative along $X$ obeys the Stratonovich equation

$$ \nabla_{\circ dX_t} \|\cdot\|_t = 0, \qquad (13.13) $$

and if $V_t = \|\cdot\|_t$ for fixed $v \in T_{X_0}M$, then $V_t$ is the parallel translate of $v$. The connection Laplacian acting on sections (the rough Laplacian) appears in Bochner-type identities and in the analysis of vector-valued heat flows:

$$ \Box \Phi = \nabla^* \nabla \Phi, \qquad (13.14) $$

and is related to the generator of the lifted diffusion on vector bundles.

**Smoothness of transition probabilities and Malliavin nondegeneracy.** Since the generator $\frac{1}{2}\Delta_g$ is elliptic (the vector fields $e_a$ span the tangent space), Hörmander's condition is satisfied and finite-dimensional Malliavin theory implies that $p_t(x, y)$ is $C^\infty$ in $(x, y)$ for $t > 0$. The Malliavin covariance matrix is invertible for $t > 0$, which yields Gaussian-type upper and lower bounds on $p_t(x, y)$ under curvature and injectivity radius control.

**Long-time behaviour and spectral theory.** On complete manifolds with suitable confinement (e.g. compact $M$ or potentials growing at infinity) the spectrum of $-\frac{1}{2}\Delta_g$ is discrete and the heat semigroup admits an eigenexpansion. On noncompact manifolds the spectral measure, existence of a spectral gap, and asymptotic behaviour of $P_t$ are governed by geometry (volume growth, curvature decay, ends structure).

**Applications and further directions.** Brownian motion on manifolds connects probabilistic representations of geometric PDEs (e.g. probabilistic proofs of the Hodge theorem, index theorems), geometric analysis (comparison theorems, heat kernel estimates), and stochastic
differential geometry (stochastic flows, stochastic development). Extensions include analysis on manifolds with boundary (reflecting Brownian motion and boundary local times), on non-Riemannian settings (sub-Riemannian Brownian motion), and interactions with gauge fields and stochastic parallel transport in vector bundles.

In summary, Brownian motion on $(M, g)$ is the diffusion process generated by $\frac{1}{2}\Delta_g$; it admits both intrinsic geometric constructions via the orthonormal frame bundle and analytical manifestations through the heat semigroup and heat kernel, with its qualitative and quantitative behavior tightly controlled by the Riemannian geometry of $M$.

## 13.2 Geometric Interpretation via Horizontal Lifts

Let $(M, g)$ be a smooth, connected Riemannian manifold of dimension $d$, and let $O(M)$ denote its orthonormal frame bundle, i.e.

$$O(M) := \bigcup_{x \in M} O_x(M), \quad O_x(M) := \{u : \mathbb{R}^d \to T_x M \text{ is an isometry w.r.t. } g_x\}. \quad (13.15)$$

The projection map $\pi : O(M) \to M$ sends an orthonormal frame $u$ to its base point $\pi(u) = x$. The tangent bundle $TO(M)$ admits a canonical decomposition into a horizontal and a vertical subbundle:

$$TO(M) = \mathcal{H} \oplus \mathcal{V}, \quad (13.16)$$

where the horizontal distribution $\mathcal{H}$ is defined via the Levi-Civita connection $\nabla$ on $(M, g)$. The horizontal lift of a tangent vector $v \in T_x M$ at $u \in O_x(M)$ is the unique vector $h_u(v) \in \mathcal{H}_u$ satisfying

$$\pi_* h_u(v) = v, \quad \omega(h_u(v)) = 0, \quad (13.17)$$

where $\omega$ denotes the connection 1-form on $O(M)$ induced by $\nabla$.

**Canonical horizontal vector fields.** Fix the canonical basis $(e_1, \dots, e_d)$ of $\mathbb{R}^d$. For each $a = 1, \dots, d$, define the canonical horizontal vector field $H_a$ on $O(M)$ by

$$H_a(u) := h_u(ue_a), \quad (13.18)$$

i.e. $H_a(u)$ is the horizontal lift of the tangent vector $ue_a \in T_{\pi(u)}M$. These vector fields form an orthonormal frame of the horizontal distribution $\mathcal{H} \subset TO(M)$.

**Stochastic development on $O(M)$**. Let $(B_t^1, \dots, B_t^d)$ be standard independent Brownian motions in $\mathbb{R}^d$. The horizontal lift of Brownian motion on $M$ is the process $(U_t)_{t \ge 0}$ in $O(M)$ satisfying the Stratonovich stochastic differential equation

$$dU_t = \sum_{a=1}^{d} H_a(U_t) \circ dB_t^a, \quad U_0 = u_0 \in O_{x_0}(M), \quad (13.19)$$

where $u_0$ is an orthonormal frame at the starting point $x_0 \in M$. This SDE is geometrically well-defined because the vector fields $H_a$ are globally smooth and horizontal, and the Stratonovich formulation respects the manifold structure of $O(M)$.
**Projection to Brownian motion on M.** Define the projected process $X_t = \pi(U_t)$. Applying Itô's formula in local coordinates and using the properties of the horizontal vector fields, one verifies that

$$ \mathcal{L}(f \circ \pi)(u) = \frac{1}{2} \sum_{a=1}^{d} H_a^2(f \circ \pi)(u) = \frac{1}{2}(\Delta_g f)(\pi(u)), \quad (13.20) $$

so that the generator of the process $X_t$ is precisely $\frac{1}{2}\Delta_g$, the Laplace-Beltrami operator. Hence $X_t$ is Brownian motion on the Riemannian manifold $(M, g)$, and the law of $X_t$ is independent of the choice of initial frame $u_0$ above $x_0$.

**Geometric interpretation.** The equation (13.19) defines a stochastic development of Euclidean Brownian motion $(B_t)$ onto the manifold $(M, g)$ via the connection-induced horizontal lift. In other words, the Euclidean noise is “rolled” along $M$ without slipping or twisting. The projection $X_t = \pi(U_t)$ depends solely on the Riemannian structure $(M, g)$ and not on any choice of coordinates.

This construction highlights the intrinsic nature of Brownian motion on Riemannian manifolds: it arises from the geometry of $(M, g)$ alone, as the projection of a horizontal diffusion on the orthonormal frame bundle governed by the Levi-Civita connection.

## 13.3 Heat Semigroup and Heat Kernel on Manifolds

Let $(M, g)$ be a smooth, connected, complete Riemannian manifold of dimension $d$, with associated volume measure $dvol_g$. The Laplace-Beltrami operator $\Delta_g$ acts on smooth functions $f \in C^\infty(M)$ as

$$ \Delta_g f = \operatorname{div}_g (\nabla_g f), \quad (13.21) $$

where $\nabla_g f$ denotes the Riemannian gradient and $\operatorname{div}_g$ the Riemannian divergence. We consider the infinitesimal generator

$$ L := \frac{1}{2} \Delta_g, \quad (13.22) $$

which is essentially self-adjoint on $L^2(M, dvol_g)$ under appropriate completeness assumptions (for instance, when $M$ is complete without boundary).

**Heat semigroup.** The operator $L = \frac{1}{2}\Delta_g$ generates a strongly continuous, symmetric contraction semigroup $(T_t)_{t \ge 0}$ on $L^2(M)$, called the *heat semigroup*, satisfying:

$$ T_t f = e^{tL} f, \quad T_0 = I, \quad T_{t+s} = T_t T_s, \quad (13.23) $$

and the limit property $\lim_{t \to 0^+} T_t f = f$ in $L^2(M)$ for all $f \in L^2(M)$.

If $X_t$ denotes the Brownian motion on $(M, g)$ starting at $x$, defined as the diffusion process with generator $\frac{1}{2}\Delta_g$, then for any bounded measurable $f$,

$$ (T_t f)(x) = \mathbb{E}^x[f(X_t)]. \quad (13.24) $$
Thus, $T_t$ describes the evolution of expectations of functionals along the Brownian motion on $M$.

**Heat kernel representation.** The heat semigroup admits an integral representation in terms of a smooth, nonnegative function $p_t(x, y)$, known as the *heat kernel*, such that

$$ (T_t f)(x) = \int_M p_t(x, y) f(y) \, d\mathrm{vol}_g(y), \qquad (13.25) $$

for all $f \in L^2(M)$ and $t > 0$. The heat kernel $p_t(x, y)$ is the *minimal fundamental solution* to the heat equation

$$ \frac{\partial u}{\partial t}(t, x) = \frac{1}{2} \Delta_g u(t, x), \quad u(0, x) = f(x), \qquad (13.26) $$

in the sense that

$$ u(t, x) = (T_t f)(x) = \int_M p_t(x, y) f(y) \, d\mathrm{vol}_g(y) \qquad (13.27) $$

is the unique minimal nonnegative solution of (13.26) for every nonnegative initial data $f$.

**Semigroup property of the heat kernel.** The kernel $p_t(x, y)$ satisfies the *Chapman-Kolmogorov equation* or *semigroup property*:

$$ p_{t+s}(x, y) = \int_M p_t(x, z) p_s(z, y) \, d\mathrm{vol}_g(z), \quad \forall t, s > 0. \qquad (13.28) $$

This follows directly from the semigroup identity $T_{t+s} = T_t T_s$ and the integral representation (13.25). Furthermore, for each fixed $t > 0$, $p_t(x, \cdot)$ is a smooth function on $M$ and symmetric:

$$ p_t(x, y) = p_t(y, x), \qquad (13.29) $$

reflecting the self-adjointness of $\frac{1}{2}\Delta_g$.

**Asymptotic expansion as $t \to 0^+$**. A cornerstone of geometric analysis is the short-time asymptotic expansion of $p_t(x, y)$, which reveals the influence of the manifold's curvature on diffusion. In a geodesically convex neighborhood where the geodesic distance $d_g(x, y)$ is smooth, one has

$$ p_t(x, y) \sim (4\pi t)^{-d/2} e^{-\frac{d_g(x,y)^2}{4t}} \sum_{k=0}^{\infty} a_k(x, y) t^k, \quad t \to 0^+, \qquad (13.30) $$

where:

* $d_g(x, y)$ is the Riemannian geodesic distance between $x$ and $y$,

* $a_k(x, y)$ are smooth coefficient functions on $M \times M$,

* $a_0(x, x) = 1$, and higher coefficients $a_k(x, y)$ encode curvature information.

For instance, near the diagonal $x = y$,

$$ a_1(x, x) = \frac{1}{6} R(x), \qquad (13.31) $$

where $R(x)$ is the scalar curvature of $(M, g)$.
This expansion generalizes the Euclidean heat kernel

$$p_t^{\mathbb{R}^d}(x, y) = (4\pi t)^{-d/2} \exp\left(-\frac{|x-y|^2}{4t}\right), \quad (13.32)$$

by replacing the Euclidean distance with the geodesic distance and incorporating geometric corrections through curvature-dependent coefficients $a_k(x, y)$.

**Geometric significance.** The heat kernel $p_t(x, y)$ encodes rich geometric and analytic information about the manifold. It determines the spectral data of the Laplace-Beltrami operator, since

$$p_t(x, y) = \sum_{j=0}^{\infty} e^{-\lambda_j t} \phi_j(x) \phi_j(y), \quad (13.33)$$

where $\{(\lambda_j, \phi_j)\}$ are the eigenpairs of $-\frac{1}{2}\Delta_g$. Moreover, heat kernel asymptotics provide deep connections between local geometry (curvature invariants) and global spectral properties (Weyl's law, index theorems, and spectral invariants).

Hence, the heat semigroup $T_t$ and its kernel $p_t(x, y)$ serve as the analytic bridge between stochastic diffusion, differential geometry, and spectral theory on Riemannian manifolds.

## 13.4 Gradient Operators and Stochastic Parallel Transport

Let $(M, g)$ be a smooth Riemannian manifold, $\nabla$ its Levi-Civita connection, and $(X_t)_{t \ge 0}$ a continuous semimartingale on $M$ (in particular a Brownian motion when its generator is $\frac{1}{2}\Delta_g$). We expand with full precision the geometric notions of gradient, covariant derivative along a semimartingale, stochastic parallel transport, and the manifold Itô formula which together furnish the analytic backbone for Malliavin calculus on manifolds.

**Covariant derivative along a curve / semimartingale.** If $\gamma: [0, T] \to M$ is a $C^1$ curve and $V(t)$ is a time-dependent vector field along $\gamma$ (i.e. $V(t) \in T_{\gamma(t)}M$), the covariant derivative of $V$ along $\gamma$ is the vector field $\nabla_{\dot{\gamma}(t)}V \in T_{\gamma(t)}M$ defined by the Levi-Civita connection. For a continuous semimartingale $X_t$ and an adapted process $V_t$ with values in $T_{X_t}M$ one defines the stochastic covariant derivative in Stratonovich form by the pathwise limit

$$\nabla_{\circ dX_t} V_t := \text{Stratonovich differential of } V_t \text{ along } X_t, \quad (13.34)$$

characterized locally (in coordinates) by replacing ordinary derivatives by covariant derivatives and ordinary differentials by Stratonovich differentials. Concretely, if $V_t = V^i(t, X_t) \partial_{x^i}$ in local coordinates, then

$$\nabla_{\circ dX_t} V_t = (dV^i(t, X_t) + \Gamma^i_{jk}(X_t)V^j(t, X_t) \circ dX_t^k) \partial_{x^i}, \quad (13.35)$$

where $\Gamma^i_{jk}$ are the Christoffel symbols of $\nabla$.

**Stochastic parallel transport.** Let $v \in T_{X_0}M$ be a fixed initial vector. The stochastic parallel transport $P_t: T_{X_0}M \to T_{X_t}M$ along the semimartingale $X_t$ is defined as the unique
family of linear isometries satisfying the covariant Stratonovich ordinary differential equation

$$ \nabla_{\circ dX_t} (P_t v) = 0, \quad P_0 = \mathrm{Id}_{T_{X_0} M}. \qquad (13.36) $$

Equivalently, $V_t := P_t v$ is the unique adapted $T_{X_t} M$-valued process such that $V_0 = v$ and the covariant derivative along $X$ vanishes in Stratonovich sense. In local coordinates this SDE becomes

$$ dV_t^i + \Gamma_{jk}^i(X_t) V_t^j \circ dX_t^k = 0, \qquad (13.37) $$

which preserves the inner product: $g_{X_t}(P_t v, P_t w) = g_{X_0}(v, w)$ for all $v, w \in T_{X_0} M$.

**Frame-bundle formulation.** Let $\pi : O(M) \to M$ be the orthonormal frame bundle and $H_a$ the canonical horizontal vector fields. If $U_t$ solves the horizontal Stratonovich SDE on $O(M)$

$$ dU_t = \sum_{a=1}^{d} H_a(U_t) \circ dB_t^a, \quad U_0 = u_0, \qquad (13.38) $$

then $X_t = \pi(U_t)$ is the stochastic development of the Euclidean Brownian motion $B_t = (B_t^1, \dots, B_t^d)$. The stochastic parallel transport $P_t$ is then given by the linear map

$$ P_t = U_t U_0^{-1} : T_{X_0} M \to T_{X_t} M, \qquad (13.39) $$

(i.e. $U_t$ carries the initial frame to the frame at time $t$). This frame-bundle picture is the canonical geometric construction used in proofs and in Malliavin calculus on manifolds.

**Itô versus Stratonovich and the geometric Itô formula.** For manifold-valued semi-martingales the Stratonovich calculus is geometrically natural because Stratonovich differentials transform covariantly under changes of charts. Let $f \in C^\infty(M)$. The Stratonovich Itô formula gives

$$ f(X_t) - f(X_0) = \int_0^t \langle \nabla f(X_s), \circ dX_s \rangle, \qquad (13.40) $$

where $\nabla f$ is the Riemannian gradient, and $\langle \nabla f, \circ dX \rangle$ denotes the Stratonovich integral of the 1-form $df$ against $X$. Converting to Itô differentials (for example when $X$ is Brownian motion) one obtains the covariant Itô formula: if $X$ is a Brownian motion with generator $\frac{1}{2}\Delta_g$, then almost surely

$$ f(X_t) = f(X_0) + \int_0^t \langle \nabla f(X_s), dX_s \rangle + \frac{1}{2} \int_0^t (\Delta_g f)(X_s) ds, \qquad (13.41) $$

where the first stochastic integral is an Itô integral (with the necessary Itô correction already encoded in the second term). Here the Laplace-Beltrami operator $\Delta_g$ appears as the trace of the second covariant derivative:

$$ \Delta_g f = \mathrm{tr} (\nabla^2 f). \qquad (13.42) $$

**Covariant Itô formula for vector fields and one-forms.** If $V$ is a $C^2$ vector field on $M$ and $V(X_t)$ denotes its evaluation along $X_t$, then the covariant Itô formula reads (in Itô form)

$$ \nabla_{dX_t} V(X_t) = \nabla_{dX_t} V = \nabla_{\circ dX_t} V - \frac{1}{2} \mathrm{Ric}^{\sharp}(V(X_t)) dt, \qquad (13.43) $$
with appropriate curvature correction terms appearing when converting from Stratonovich to Itô; the appearance of Ricci-type terms depends on whether one considers horizontal lifts or the Itô development (for Brownian motion the precise correction yields the generator $\frac{1}{2}\Delta_g$). For one-forms $\alpha$ the covariant Itô formula gives the evolution of $\alpha(X_t)$ along the noise with the connection Laplacian acting in the drift.

**Linearization and derivative flow.** Let $\Phi_{s,t}: M \to M$ denote the stochastic flow map (when it exists) associated to the SDE defining $X$. The derivative $D\Phi_{0,t}(x): T_xM \to T_{X_t}M$ satisfies a linear covariant stochastic differential equation; in parallel-transported coordinates one obtains the linear Stratonovich SDE

$$ \nabla_{\circ dX_t} Y_t = (\nabla Y)(X_t) \circ dX_t, \qquad (13.44) $$

and if $X_t$ is the solution driven by frame noise $e_a \circ dB_t^a$, the linearization becomes

$$ \nabla_{\circ dX_t} Y_t = \sum_{a=1}^{d} \nabla_{Y_t} e_a(X_t) \circ dB_t^a. \qquad (13.45) $$

Writing the linearization in the frame bundle and projecting yields an SDE for $D\Phi_{0,t}$ whose solution can be expressed via stochastic parallel transport and iterated covariant derivatives of the driving vector fields. This derivative flow is the fundamental object in the Malliavin differentiation of $X_t$.

**Malliavin derivative and representation via parallel transport.** In the manifold setting Malliavin differentiation of the solution $X_t$ to a stochastic differential equation driven by an $\mathbb{R}^d$-Brownian motion admits a geometric representation: for $0 \le s \le t$,

$$ D_s X_t = P_t \circ P_s^{-1} \sigma(X_s) 1_{[0,t]}(s), \qquad (13.46) $$

where $\sigma(X_s): \mathbb{R}^d \to T_{X_s}M$ is the map sending the canonical Euclidean noise directions to the corresponding tangent vectors (for Brownian motion $\sigma$ is realized by the frame fields $e_a$), and $P_t, P_s$ are stochastic parallel transports along $X$. This formula is to be interpreted in local charts or via the frame bundle; it expresses the infinitesimal sensitivity of $X_t$ to perturbations of the driving Brownian path at time $s$ by transporting the perturbation forward along the path.

Consequently, the finite-dimensional Malliavin covariance of a projection $\ell(X_t)$ (with $\ell: T_{X_t}M \to \mathbb{R}^m$ linear) can be written in terms of parallel transports:

$$ \sigma_{\ell(X_t)} = \int_0^t (\ell \circ P_t \circ P_s^{-1} \circ \sigma(X_s)) (\ell \circ P_t \circ P_s^{-1} \circ \sigma(X_s))^T ds. \qquad (13.47) $$

Nondegeneracy of this matrix (in the Bouleau-Hirsch sense) yields absolute continuity and smoothness of the law of $\ell(X_t)$.

**Conclusion and role in Malliavin calculus.** The geometric constructs above—covariant derivative, stochastic parallel transport, frame-bundle horizontal lift, and the covariant Itô formula—provide the necessary structure to formulate Malliavin calculus intrinsically on manifolds.