Stochastic parallel transport furnishes canonical identifications of tangent spaces along sample paths, enabling one to express Malliavin derivatives, covariance operators, and integration-by-parts formulae in a coordinate-free manner. These tools yield hypoellipticity criteria (Hörmander-type conditions carried out on vector fields on $M$), smoothness of transition densities, and precise gradient estimates for the heat semigroup on manifolds, all central to stochastic analysis in geometric settings.

## 13.5 Bochner Identity and Curvature Effects

Let $(M, g)$ be a smooth, connected, $d$-dimensional Riemannian manifold equipped with its Levi-Civita connection $\nabla$, Riemann curvature tensor $R$, and Ricci curvature tensor $Ric$. For $f \in C^\infty(M)$, we denote by $\nabla f$ its Riemannian gradient, $\nabla^2 f = \nabla(\nabla f)$ its covariant Hessian, and $\Delta_g f = \text{div}(\nabla f)$ the Laplace-Beltrami operator. The Bochner-Weitzenböck identity rigorously describes the interplay between the second covariant derivatives of $f$, the Laplacian, and the Ricci curvature.

### 13.5.1 Preliminary definitions

**Gradient and Hessian.** For any smooth function $f \in C^\infty(M)$, the Riemannian gradient $\nabla f$ is the vector field defined by

$$g(\nabla f, X) = df(X), \quad \forall X \in \Gamma(TM), \qquad (13.48)$$

and the covariant Hessian $\nabla^2 f$ is the symmetric $(0,2)$-tensor given by

$$\nabla^2 f(X, Y) = g(\nabla_X \nabla f, Y) = X(Y(f)) - (\nabla_X Y)(f), \qquad (13.49)$$

for all vector fields $X, Y$. Its squared norm is

$$\|\nabla^2 f\|^2 = g^{ia}g^{jb}(\nabla_i \nabla_j f)(\nabla_a \nabla_b f), \qquad (13.50)$$

independent of the choice of coordinates.

**Laplacian and Divergence.** The Laplace-Beltrami operator acts as

$$\Delta_g f = \text{div}(\nabla f) = g^{ij} \nabla_i \nabla_j f = \frac{1}{\sqrt{|g|}} \partial_i \left( \sqrt{|g|} g^{ij} \partial_j f \right), \qquad (13.51)$$

where $|g| = \det(g_{ij})$. It is self-adjoint on $L^2(M, \text{vol}_g)$ and coincides with the generator of Brownian motion on $M$.

### 13.5.2 Derivation of the Bochner-Weitzenböck identity

Let $f \in C^\infty(M)$. We compute the Laplacian of the squared gradient norm $u = |\nabla f|^2 = g(\nabla f, \nabla f)$. Differentiating covariantly,

$$\nabla_i u = 2 g(\nabla_i \nabla f, \nabla f) = 2 (\nabla_i \nabla_j f)(\nabla^j f), \qquad (13.52)$$
and hence

$$ \Delta_g u = \nabla^i \nabla_i u = 2 \nabla^i ((\nabla_i \nabla_j f)(\nabla^j f)) = 2 (\nabla^i \nabla_i \nabla_j f)(\nabla^j f) + 2 (\nabla_i \nabla_j f)(\nabla^i \nabla^j f). \quad (13.53) $$

Now, the first term can be rewritten by the curvature commutation formula:

$$ \nabla^i \nabla_i \nabla_j f = \nabla_j \nabla^i \nabla_i f - R_j^k \nabla_k f = \nabla_j (\Delta_g f) - \mathrm{Ric}_{jk} \nabla^k f, \quad (13.54) $$

where we used the Ricci identity

$$ (\nabla_i \nabla_j - \nabla_j \nabla_i) \nabla_k f = R_{ijk}^k \nabla_l f \quad \text{and hence} \quad R_{jk} = R_{ijk}^i. \quad (13.55) $$

Substituting this back, we obtain

$$ \frac{1}{2} \Delta_g |\nabla f|^2 = (\nabla_j f)(\nabla^j \Delta_g f) + (\nabla_i \nabla_j f)(\nabla^i \nabla^j f) + \mathrm{Ric}_{jk} (\nabla^j f)(\nabla^k f). \quad (13.56) $$

In invariant tensor notation, this reads

$$ \frac{1}{2} \Delta_g |\nabla f|^2 = \langle \nabla f, \nabla (\Delta_g f) \rangle + \|\nabla^2 f\|^2 + \mathrm{Ric}(\nabla f, \nabla f), \quad (13.57) $$

which is the Bochner-Weitzenböck identity.

### 13.5.3 Interpretations and geometric consequences

**Analytic interpretation.** The term $\|\nabla^2 f\|^2$ measures the local convexity of $f$, while $\mathrm{Ric}(\nabla f, \nabla f)$ captures how curvature affects the diffusion of gradients along the manifold. Equation (13.57) thus relates second-order derivatives of $f$ and the geometry of $M$.

**Integration formula.** Integrating (13.57) over $M$ with respect to the Riemannian volume measure and using the self-adjointness of $\Delta_g$, we obtain

$$ \int_M \|\nabla^2 f\|^2 d\mathrm{vol}_g + \int_M \mathrm{Ric}(\nabla f, \nabla f) d\mathrm{vol}_g = \int_M (\Delta_g f)^2 d\mathrm{vol}_g. \quad (13.58) $$

This equality is a central tool in elliptic regularity and spectral geometry, expressing the $L^2$-norm of the Laplacian in terms of Hessian and curvature contributions.

**Probabilistic interpretation.** If $(X_t)$ is Brownian motion on $M$, Itô's formula and (13.57) yield

$$ d|\nabla f(X_t)|^2 = (\Delta_g |\nabla f|^2)(X_t) dt + \text{martingale term}, \quad (13.59) $$

so that curvature affects the expected growth of $|\nabla f(X_t)|^2$. When $\mathrm{Ric} \ge 0$, the expectation $\mathbb{E}[|\nabla f(X_t)|^2]$ is subharmonic, providing a stochastic comparison principle.

**Bakry-Émery curvature criterion.** Define the carré du champ operator $\Gamma(f) = |\nabla f|^2$ and its iterated form

$$ \Gamma_2(f) = \frac{1}{2} (\Delta_g \Gamma(f) - 2\Gamma(f, \Delta_g f)), \quad (13.60) $$
where $\Gamma(f,g) = \langle \nabla f, \nabla g \rangle$. By the Bochner identity,

$$ \Gamma_2(f) = \|\nabla^2 f\|^2 + \mathrm{Ric}(\nabla f, \nabla f). \qquad (13.61) $$

A lower bound $\mathrm{Ric} \ge Kg$ is thus equivalent to the Bakry-Émery curvature-dimension condition $CD(K, \infty)$:

$$ \Gamma_2(f) \ge K \Gamma(f), \qquad (13.62) $$

which underpins gradient estimates, logarithmic Sobolev inequalities, and exponential convergence to equilibrium for diffusion semigroups.

**Conclusion.** The Bochner-Weitzenböck identity provides a precise differential relation between curvature and the behavior of gradients and Laplacians. It lies at the intersection of Riemannian geometry, stochastic analysis, and functional inequalities, serving as the foundational analytic identity behind many curvature-dimension and diffusion semigroup results.

## 13.6 Diffusions with Drift and Divergence Form Operators

Let $(M, g)$ be a smooth, connected, complete Riemannian manifold of dimension $d$, and let $b \in \Gamma(TM)$ be a smooth vector field on $M$, referred to as the *drift field*. We define the second-order differential operator

$$ Lf = \frac{1}{2} \Delta_g f + \langle b, \nabla f \rangle, \qquad (13.63) $$

for all $f \in C^\infty(M)$, where $\Delta_g$ denotes the Laplace-Beltrami operator and $\nabla f$ is the Riemannian gradient of $f$. This operator $L$ is called the *generator* of a diffusion process with drift $b$.

### 13.6.1 Stochastic Differential Representation

Let $\{B_t\}_{t \ge 0}$ denote a standard Brownian motion in $\mathbb{R}^d$, and let $\{e_a(x)\}_{a=1}^d$ be a local orthonormal frame field on $M$, i.e.

$$ g(e_a(x), e_b(x)) = \delta_{ab}, \quad \forall x \in M. \qquad (13.64) $$

The diffusion process $\{X_t\}_{t \ge 0}$ generated by $L$ satisfies the Stratonovich stochastic differential equation (SDE)

$$ dX_t = b(X_t) dt + \sum_{a=1}^{d} e_a(X_t) \circ dB_t^a, \qquad (13.65) $$

where the stochastic integral is taken in the Stratonovich sense to preserve the coordinate-invariance under changes of local charts. Equation (13.65) defines a stochastic flow on $M$ whose infinitesimal generator coincides with $L$ as defined in (13.63).

The equivalence between the SDE and the generator can be rigorously verified using Itô's formula on manifolds: for any $f \in C^\infty(M)$,

$$ df(X_t) = \langle \nabla f(X_t), b(X_t) \rangle dt + \frac{1}{2} (\Delta_g f)(X_t) dt + \sum_{a=1}^{d} \langle \nabla f(X_t), e_a(X_t) \rangle \circ dB_t^a. \qquad (13.66) $$
Taking expectations and differentiating under the integral sign gives

$$ \frac{d}{dt} \mathbb{E}[f(X_t)] = \mathbb{E}[Lf(X_t)], \qquad (13.67) $$

confirming that $L$ is indeed the infinitesimal generator of the diffusion.

### 13.6.2 Invariant Measures and Reversibility

Let $\mu$ be a smooth measure on $M$ absolutely continuous with respect to the Riemannian volume form $dvol_g$, say

$$ d\mu = \rho(x) dvol_g(x), \qquad (13.68) $$

where $\rho: M \to (0, \infty)$ is a smooth density function. We say that $\mu$ is invariant under the diffusion generated by $L$ if, for all $f, g \in C_c^\infty(M)$,

$$ \int_M (Lf) g \, d\mu = \int_M f (L^*g) \, d\mu, \qquad (13.69) $$

where $L^*$ is the formal adjoint of $L$ in $L^2(M, \mu)$. A sufficient condition for invariance is the divergence-free condition in weighted form:

$$ \mathrm{div}_{\mu}(b) = \mathrm{div}(b) + \langle b, \nabla \log \rho \rangle = 0, \qquad (13.70) $$

where $\mathrm{div}$ denotes the Riemannian divergence.

### 13.6.3 Gradient Drift and Self-Adjoint Realization

An important special case arises when the drift field is a gradient, i.e.

$$ b = \nabla V \qquad (13.71) $$

for some smooth potential function $V: M \to \mathbb{R}$. Then the operator (13.63) becomes

$$ Lf = \frac{1}{2} \Delta_g f + \langle \nabla V, \nabla f \rangle. \qquad (13.72) $$

We define the weighted measure

$$ d\mu_V = e^{-2V(x)} dvol_g(x). \qquad (13.73) $$

A direct computation using the divergence theorem on Riemannian manifolds yields

$$ \int_M (Lf) g \, d\mu_V = \frac{1}{2} \int_M \langle \nabla f, \nabla g \rangle \, d\mu_V = \int_M f (Lg) \, d\mu_V, $$

for all compactly supported smooth functions $f, g \in C_c^\infty(M)$. Hence $L$ is symmetric (and closable) in $L^2(M, \mu_V)$.

The operator can equivalently be expressed in divergence form:

$$ Lf = \frac{1}{2} e^{2V} \mathrm{div}(e^{-2V} \nabla f). \qquad (13.74) $$
This expression emphasizes the geometric structure of the diffusion: it is a self-adjoint elliptic operator on the weighted Riemannian space $(M, g, \mu_V)$, known as a *weighted Laplacian* or *Witten Laplacian*.

### 13.6.4 Dirichlet Form Representation

The corresponding Dirichlet form $(\mathcal{E}, \mathcal{F})$ associated with $L$ on $L^2(M, \mu_V)$ is given by

$$ \mathcal{E}(f, g) = \frac{1}{2} \int_M \langle \nabla f, \nabla g \rangle d\mu_V, \qquad (13.75) $$

with domain $\mathcal{F} = H^1(M, \mu_V)$, the Sobolev space of square-integrable functions with square-integrable gradients. This form is symmetric, closed, and Markovian, implying that $L$ is the unique self-adjoint operator associated with $\mathcal{E}$.

### 13.6.5 Summary of the Analytic and Probabilistic Correspondence

The analytic and probabilistic viewpoints are summarized as follows:

<table><thead><tr><th>Concept</th><th>Analytic Representation</th><th>Probabilistic Interpretation</th></tr></thead><tbody><tr><td>Generator</td><td>$L = \frac{1}{2}\Delta_g + \langle b, \nabla \rangle$</td><td>Drift-diffusion dynamics</td></tr><tr><td>Adjoint</td><td>$L^* = \frac{1}{2}\Delta_g - \text{div}(b\cdot)$</td><td>Backward Kolmogorov operator</td></tr><tr><td>Invariant measure</td><td>$\text{div}_\mu(b) = 0$</td><td>Stationary law of diffusion</td></tr><tr><td>Gradient drift</td><td>$b = \nabla V$</td><td>Reversible diffusion</td></tr><tr><td>Weighted Laplacian</td><td>$L = \frac{1}{2}e^{2V}\text{div}(e^{-2V}\nabla f)$</td><td>Self-adjoint generator in $L^2(M, e^{-2V}d\text{vol}_g)$</td></tr></tbody></table>

This framework provides a mathematically rigorous unification of Riemannian geometry, stochastic analysis, and functional analysis through the study of elliptic operators in divergence form and their associated diffusion processes.

## 13.7 Geometric Dirichlet Forms

Let $(M, g)$ be a smooth, connected, complete Riemannian manifold of dimension $d$, equipped with the Riemannian metric tensor $g = (g_{ij})$ and associated volume measure $d\text{vol}_g = \sqrt{|g|} dx^1 \cdots dx^d$, where $|g| = \det(g_{ij})$. We denote by $\nabla f$ the Riemannian gradient of a smooth function $f \in C^\infty(M)$ and by $\Delta_g$ the Laplace-Beltrami operator acting on scalar functions.

### 13.7.1 Definition and Domain of the Dirichlet Form

The geometric Dirichlet form associated with the Riemannian structure $(M, g)$ is defined by

$$ \mathcal{E}(f, g) = \frac{1}{2} \int_M \langle \nabla f, \nabla g \rangle_g d\text{vol}_g, \qquad (13.76) $$

for all $f, g \in C_c^\infty(M)$, where $\langle \cdot, \cdot \rangle_g$ denotes the inner product on the tangent bundle $TM$ induced by $g$. The natural domain of $\mathcal{E}$ is the first-order Sobolev space

$$ \mathcal{D}(\mathcal{E}) = H^1(M) = \{f \in L^2(M, d\text{vol}_g) \mid \nabla f \in L^2(TM, d\text{vol}_g)\}, \qquad (13.77) $$
equipped with the norm

$$ \|f\|_{H^1(M)}^2 = \|f\|_{L^2(M)}^2 + \|\nabla f\|_{L^2(TM)}^2. \qquad (13.78) $$

### 13.7.2 Fundamental Analytic Properties

(i) **Symmetry.** By definition of the Riemannian inner product, $\mathcal{E}(f,g) = \mathcal{E}(g,f)$ for all $f,g \in \mathcal{D}(\mathcal{E})$.

(ii) **Positivity and Markovianity.** For any $f \in \mathcal{D}(\mathcal{E})$, one has $\mathcal{E}(f,f) \ge 0$. Moreover, if $\eta: \mathbb{R} \to \mathbb{R}$ is a normal contraction (i.e. Lipschitz with $|\eta(r) - \eta(s)| \le |r - s|$), then

$$ \mathcal{E}(\eta \circ f, \eta \circ f) \le \mathcal{E}(f, f), \qquad (13.79) $$

which ensures that $\mathcal{E}$ is a *Dirichlet form* in the sense of Fukushima-Oshima-Takeda.

(iii) **Closability.** Since $C_c^\infty(M)$ is dense in $H^1(M)$ with respect to the $H^1$ norm, the bilinear form $\mathcal{E}$ is closable on $L^2(M, dvol_g)$, and its closure has domain exactly $H^1(M)$.

(iv) **Strong locality.** If $f,g \in \mathcal{D}(\mathcal{E})$ have compact supports and $g$ is constant on a neighborhood of supp($f$), then

$$ \mathcal{E}(f,g) = 0. \qquad (13.80) $$

This *strong locality* property characterizes the geometric nature of the form—it depends only on the local behavior of the gradients.

### 13.7.3 Associated Self-Adjoint Operator

By the general theory of closed, symmetric, and strongly local Dirichlet forms (see Fukushima-Oshima-Takeda, *Dirichlet Forms and Symmetric Markov Processes*), there exists a unique nonpositive self-adjoint operator $L$ on $L^2(M, dvol_g)$ such that

$$ \mathcal{E}(f,g) = -\langle Lf, g \rangle_{L^2(M)}, \quad \forall f \in \mathcal{D}(L), g \in \mathcal{D}(\mathcal{E}). \qquad (13.81) $$

For the geometric Dirichlet form (13.76), this operator $L$ is precisely the Riemannian Laplacian:

$$ L = \frac{1}{2} \Delta_g. \qquad (13.82) $$

The domain $\mathcal{D}(L)$ consists of those $f \in H^1(M)$ for which $\Delta_g f \in L^2(M)$, i.e.

$$ \mathcal{D}(L) = \{f \in H^1(M) : \Delta_g f \in L^2(M)\}. \qquad (13.83) $$

### 13.7.4 Functional Analytic Representation

The spectral theorem for unbounded self-adjoint operators implies that the energy form (13.76) can be expressed in operator-theoretic terms as

$$ \mathcal{E}(f, f) = \|(-L)^{1/2} f\|_{L^2(M)}^2, \qquad (13.84) $$
for all $f \in \mathcal{D}(\mathcal{E})$. Here $(-L)^{1/2}$ denotes the square-root of the nonpositive self-adjoint operator $-L$ obtained through functional calculus. Equation (13.84) rigorously identifies the Dirichlet form as the quadratic form of the operator $\frac{1}{2}\Delta_g$.

### 13.7.5 Geometric and Probabilistic Interpretations

(a) **Geometric aspect.** The energy $\mathcal{E}(f, f)$ represents the total Dirichlet energy of $f$ on $(M, g)$:

$$ \mathcal{E}(f, f) = \frac{1}{2} \int_M |\nabla f|^2 d\text{vol}_g, \qquad (13.85) $$

which measures the average rate of change of $f$ with respect to the metric $g$.

(b) **Probabilistic aspect.** The Markov semigroup $\{T_t\}_{t \ge 0}$ associated with $L = \frac{1}{2}\Delta_g$ is given by

$$ T_t f(x) = \mathbb{E}^x[f(X_t)], \qquad (13.86) $$

where $\{X_t\}_{t \ge 0}$ is the Brownian motion on $(M, g)$. Thus, $\mathcal{E}$, $\Delta_g$, and Brownian motion are connected through the analytic-geometric-probabilistic correspondence:

$$ \text{Dirichlet form } \mathcal{E} \iff \text{Laplacian } \frac{1}{2}\Delta_g \iff \text{Brownian motion on } (M, g). \qquad (13.87) $$

### 13.7.6 Summary

The geometric Dirichlet form provides a unified framework that simultaneously encodes:

* the analytic structure of the Laplace-Beltrami operator as a self-adjoint elliptic operator on $L^2(M, d\text{vol}_g)$,

* the geometric structure of the manifold through the Riemannian metric tensor $g$, and

* the probabilistic structure of Brownian motion as the Markov process associated with the form.

Hence, the equality

$$ \mathcal{E}(f, f) = \|(-L)^{1/2} f\|_{L^2(M)}^2 \qquad (13.88) $$

serves as the rigorous bridge between geometry, analysis, and stochastic processes on Riemannian manifolds.

## 13.8 Feynman-Kac Formula on Manifolds

Let $(M, g)$ be a smooth, connected, complete Riemannian manifold of dimension $d$, with Riemannian metric tensor $g = (g_{ij})$ and associated volume measure $d\text{vol}_g = \sqrt{|g|} dx^1 \cdots dx^d$. Denote by $\Delta_g$ the Laplace-Beltrami operator acting on $C^\infty(M)$, defined locally in coordinates by

$$ \Delta_g f = \frac{1}{\sqrt{|g|}} \partial_i \left( \sqrt{|g|} g^{ij} \partial_j f \right), \qquad (13.89) $$

and by $\{X_t\}_{t \ge 0}$ the Brownian motion on $(M, g)$ generated by $\frac{1}{2}\Delta_g$.
### 13.8.1 The Schrödinger-type Operator

Let $V: M \to \mathbb{R}$ be a measurable potential function satisfying

$$V(x) \geq -C_V, \quad \text{for some constant } C_V \geq 0, \qquad (13.90)$$

so that $V$ is bounded from below. Define the Schrödinger-type operator

$$H = -\frac{1}{2}\Delta_g + V, \qquad (13.91)$$

as an unbounded self-adjoint operator on $L^2(M, dvol_g)$ with domain

$$\mathcal{D}(H) = \{f \in H^2(M) \mid Vf \in L^2(M)\}. \qquad (13.92)$$

The corresponding semigroup $\{e^{-tH}\}_{t \geq 0}$ is called the *Schrödinger semigroup*, acting on $L^2(M)$ by

$$u(t, x) = (e^{-tH}f)(x), \quad u(0, x) = f(x). \qquad (13.93)$$

Formally, $u(t, x)$ satisfies the *heat equation with potential*:

$$\frac{\partial u}{\partial t} = \frac{1}{2}\Delta_g u - V(x)u. \qquad (13.94)$$

### 13.8.2 The Feynman-Kac Representation

The *Feynman-Kac formula* provides a stochastic representation of the semigroup $e^{-tH}$ in terms of Brownian motion on $(M, g)$.

**Theorem 13.1** (Feynman-Kac Formula on a Riemannian Manifold). Let $f \in L^2(M, dvol_g)$ and $V: M \to \mathbb{R}$ be a measurable potential bounded from below. Let $\{X_t\}_{t \geq 0}$ denote the Brownian motion on $(M, g)$ starting at $x \in M$, with generator $\frac{1}{2}\Delta_g$. Then, for all $t > 0$ and almost every $x \in M$,

$$(e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (13.95)$$

*Proof:* Throughout the proof $(M, g)$ is a smooth, connected, complete Riemannian manifold and $dvol_g$ its Riemannian volume measure. We denote by $X = (X_t)_{t \geq 0}$ the Brownian motion on $M$ (with generator $\frac{1}{2}\Delta_g$) constructed e.g. by stochastic development on the orthonormal frame bundle; its law $\mathbb{P}^x$ when started at $x \in M$ satisfies the usual strong Markov and martingale properties. Let $V: M \to \mathbb{R}$ be measurable and bounded from below: there exists $C \geq 0$ such that $V(x) \geq -C$ for all $x \in M$. Define the Schrödinger-type operator

$$H = -\frac{1}{2}\Delta_g + V \qquad (13.96)$$

as the self-adjoint operator obtained from the closed quadratic form

$$\mathcal{E}^V(f, f) = \frac{1}{2} \int_M |\nabla f|^2 dvol_g + \int_M V f^2 dvol_g, \quad f \in H^1(M) \cap L^2(M, |V| dvol_g), \qquad (13.97)$$

via the Friedrichs construction (this is standard since $V$ is lower bounded). The semigroup $e^{-tH}$ is thus well-defined, positivity preserving and strongly continuous on $L^2(M)$.
We must prove that for every $f \in L^2(M)$ and every $t > 0$ the identity

$$ (e^{-tH}f)(x) = \mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)] \quad (13.98) $$

holds for almost every $x \in M$.

The proof proceeds in two stages: first treat the case of bounded (and measurable) potentials, then pass to the general lower-bounded case by a monotone approximation of the potential and form convergence.

**Step 1: Bounded potential.** Assume $V \in L^\infty(M)$. Fix $f \in C_c^\infty(M)$ (a core for the Laplace-Beltrami operator) and define for $t \ge 0$

$$ u(t, x) := \mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)]. \quad (13.99) $$

We first show that $u(t, \cdot)$ belongs to $L^2(M)$ for every $t$ and that $u$ satisfies the backward Cauchy problem associated to $\partial_t - (\frac{1}{2}\Delta_g - V)$.

Since $V$ is bounded from below and above, $e^{-\int_0^t V(X_s) ds}$ is uniformly bounded by $e^{\|V^-\|_\infty t}$ and hence $u(t, x)$ is finite for all $x$ and $t$. By dominated convergence and the Feller/Markov regularity of Brownian motion one checks $u(\cdot, \cdot)$ is continuous in $(t, x)$ and $u(t, \cdot) \in C_b(M)$ for each $t > 0$.

Apply the manifold Itô formula to the Stratonovich SDE representation of Brownian motion (or equivalently to $X_t$ and the smooth test function $f$), and observe that for $s \in [0, t]$ the process

$$ M_s := e^{-\int_0^s V(X_r) dr} f(X_s) - f(X_0) - \int_0^s e^{-\int_0^r V(X_\rho) d\rho} (\frac{1}{2}\Delta_g f - Vf)(X_r) dr \quad (13.100) $$

is a martingale under $\mathbb{P}^x$. This follows from the usual Itô formula on manifolds (Stratonovich-to-Itô correction yields $\frac{1}{2}\Delta_g$ in the drift) combined with standard localization (use compact support of $f$ and boundedness of $V$). Taking expectations and evaluating at time $t$ yields

$$ u(t, x) - f(x) = \int_0^t \mathbb{E}^x[e^{-\int_0^s V(X_r) dr} (\frac{1}{2}\Delta_g f - Vf)(X_s)] ds. \quad (13.101) $$

Differentiating in $t$ (justified by dominated convergence because of the uniform bound on the exponential factor) gives, for each $t > 0$ and $x \in M$,

$$ \partial_t u(t, x) = \mathbb{E}^x[e^{-\int_0^t V(X_s) ds} (\frac{1}{2}\Delta_g f - Vf)(X_t)]. \quad (13.102) $$

But the right-hand side equals $(\frac{1}{2}\Delta_g - V)u(t, \cdot)$ evaluated at $x$ in the weak sense. Indeed, testing against a smooth compactly supported $\varphi$ and using Fubini and the Markov property justifies the interchange and shows $u$ satisfies the backward Kolmogorov equation

$$ \partial_t u(t, \cdot) = (\frac{1}{2}\Delta_g - V)u(t, \cdot), \quad u(0, \cdot) = f. \quad (13.103) $$
By semigroup theory (uniqueness of solutions to the abstract Cauchy problem in $L^2(M)$ for the generator $\frac{1}{2}\Delta_g - V$ when $V$ is bounded), we conclude that for each $t \ge 0$,

$$u(t, \cdot) = e^{-tH} f(\cdot) \qquad (13.104)$$

as elements of $L^2(M)$, whence the identity (13.98) holds for almost every $x \in M$.

**Step 2: General $V$ bounded below.** Let $V : M \to \mathbb{R}$ satisfy $V \ge -C$. Define an increasing sequence of bounded measurable potentials

$$V_n(x) := \min\{V(x), n\}, \quad n \in \mathbb{N}. \qquad (13.105)$$

Each $V_n$ is bounded (and measurable), $V_n \uparrow V$ pointwise as $n \to \infty$, and $V_n \ge -C$. For each $n$ define the closed form

$$\mathcal{E}^{(n)}(f, f) := \frac{1}{2} \int_M |\nabla f|^2 d\mathrm{vol}_g + \int_M V_n f^2 d\mathrm{vol}_g, \quad \mathcal{D}(\mathcal{E}^{(n)}) = H^1(M) \cap L^2(M, V_n d\mathrm{vol}_g), \qquad (13.106)$$

and let $H_n$ be the associated self-adjoint operator (Friedrichs extension). The corresponding semigroups $e^{-tH_n}$ are positivity preserving and by Step 1 satisfy the Feynman-Kac identity

$$(e^{-tH_n} f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V_n(X_s) ds} f(X_t) \right], \quad \text{for a.e. } x, \qquad (13.107)$$

for all $f \in L^2(M)$.

We now pass to the limit $n \to \infty$. On the analytic side, the quadratic forms $\mathcal{E}^{(n)}$ increase pointwise to $\mathcal{E}^V$ (i.e. $\mathcal{E}^{(n)}(f, f) \uparrow \mathcal{E}^V(f, f)$ for all $f$ in the common domain). By the monotone convergence theorem for closed forms (see Kato; monotone convergence of forms / Simon — standard in form methods), the operators $H_n$ converge to $H$ in the strong resolvent sense. Consequently, for each $t > 0$ and $f \in L^2(M)$,

$$e^{-tH_n} f \xrightarrow{n \to \infty} e^{-tH} f. \qquad (13.108)$$

By passing to a subsequence if necessary we may assume pointwise almost-everywhere convergence:

$$(e^{-tH_n} f)(x) \to (e^{-tH} f)(x) \quad \text{for a.e. } x \in M. \qquad (13.109)$$

On the probabilistic side, note that for every fixed path of $X$ we have the monotone pointwise convergence

$$\exp\left(-\int_{0}^{t} V_{n}(X_{s}) ds\right) \downarrow \exp\left(-\int_{0}^{t} V(X_{s}) ds\right), \quad n \to \infty, \qquad (13.110)$$

since $V_n \uparrow V$. Moreover, because $V \ge -C$ we have the uniform bound

$$\exp\left(-\int_{0}^{t} V_{n}(X_{s}) ds\right) \le \exp(Ct), \qquad (13.111)$$

so the dominated convergence theorem applies. Hence, for every bounded measurable $f$,

$$\mathbb{E}^x \left[ e^{-\int_0^t V_n(X_s) ds} f(X_t) \right] \to \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad n \to \infty, \qquad (13.112)$$
and the convergence is dominated by $e^{Ct} \mathbb{E}^x[|f(X_t)|]$.

Combining the two convergences (analytic strong $L^2$-convergence of $e^{-tH_n}f$ to $e^{-tH}f$ and dominated convergence of the probabilistic integrals), and using that the Feynman-Kac identity holds for each $n$, we conclude that for every bounded measurable $f$ and for almost every $x \in M$,

$$ (e^{-tH}f)(x) = \lim_{n \to \infty} (e^{-tH_n}f)(x) = \lim_{n \to \infty} \mathbb{E}^x \left[ e^{-\int_0^t V_n(X_s) ds} f(X_t) \right] = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right]. \quad (13.113) $$

The identity for general $f \in L^2(M)$ follows by approximation of $f$ in $L^2$ by bounded compactly supported functions and using the contractivity of the semigroups and dominated convergence on the probabilistic side.

**Conclusion:** The two-step argument establishes (13.98) for all $f \in L^2(M)$ and all $t > 0$, with the equality holding for almost every $x \in M$. This completes the rigorous proof of the Feynman-Kac formula on the Riemannian manifold $(M, g)$. $\square$

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

### 13.8.3 Analytic Consequences

**Semigroup representation:** Let the notation and hypotheses be those of Theorem 13.1: $(M, g)$ a smooth complete Riemannian manifold, $X = (X_t)_{t \ge 0}$ Brownian motion on $M$ with law $\mathbb{P}^x$ and transition kernel $p_t(x, y)$ (heat kernel for $\frac{1}{2}\Delta_g$), $V: M \to \mathbb{R}$ measurable and bounded from below, and $H = -\frac{1}{2}\Delta_g + V$ the associated self-adjoint Schrödinger operator. For fixed $t > 0$ define, for each Borel set $A \subset M$,

$$ \mu_x^t(A) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \mathbf{1}_{\{X_t \in A\}} \right]. \quad (13.114) $$

Clearly $\mu_x^t$ is a finite nonnegative measure on $M$ and

$$ \mu_x^t(M) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \right] \le e^{Ct} < \infty, \quad (13.115) $$

where $C \ge 0$ satisfies $V \ge -C$.

We claim that $\mu_x^t$ is absolutely continuous with respect to the Riemannian volume measure $dvol_g$ and therefore admits a density $p_t^V(x, \cdot) \in L_{\text{loc}}^1(M)$ (defined for a.e. $y$). To see this write, by disintegration with respect to the value $X_t$, for any nonnegative measurable function $\varphi$,

$$ \begin{aligned} \mu_x^t(\varphi) &:= \int_M \varphi(y) \mu_x^t(dy) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \varphi(X_t) \right] \\ &= \int_M \varphi(y) \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \mid X_t = y \right] p_t(x, y) dvol_g(y), \end{aligned} \quad (13.116) $$

where the second equality is the law of total expectation together with the existence of the transition density $p_t(x, y)$ for Brownian motion (classical heat kernel on a Riemannian manifold) and the existence of regular conditional distributions (or equivalently the disintegration theorem). The conditional expectation

$$ R_t(x, y) := \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \mid X_t = y \right] \quad (13.117) $$
is well-defined for $p_t(x, y) \, dvol_g$-a.e. $y$ and is finite because of the lower bound on $V$ and (13.115). From (13.116) we obtain the Radon-Nikodym derivative

$$p_t^V(x, y) := R_t(x, y) p_t(x, y), \qquad (13.118)$$

defined for $dvol_g$-a.e. $y$. Thus $\mu_x^t(dy) = p_t^V(x, y) \, dvol_g(y)$, proving absolute continuity and the existence of the density $p_t^V(x, y)$.

The representation (13.118) admits the probabilistically transparent interpretation

$$p_t^V(x, y) = p_t(x, y) \mathbb{E}_t^{x \to y} \left[ e^{-\int_0^t V(X_s) \, ds} \right], \qquad (13.119)$$

where $\mathbb{E}_t^{x \to y}[\cdot]$ denotes expectation under the pinned (Brownian) bridge measure from $x$ to $y$ over time interval $[0, t]$; the bridge expectation is precisely the regular conditional expectation appearing in (13.116). In particular the right-hand side of (13.119) is finite for $p_t(x, y) \, dvol_g$-a.e. $y$ because the bridge expectation is finite for each bridge (bounded by $e^{Ct}$).

With the kernel $p_t^V(x, y)$ thus constructed, the Feynman-Kac representation (13.95) immediately yields an integral representation for the semigroup acting on bounded measurable $f$:

$$\mathbb{E}^x \left[ e^{-\int_0^t V(X_s) \, ds} f(X_t) \right] = \int_M f(y) \mu_x^t(dy) \\ = \int_M f(y) p_t^V(x, y) \, dvol_g(y). \qquad (13.120)$$

On the analytic side, by Theorem 13.1 we have for each $f \in L^2(M)$ and every $t > 0$ the equality (for a.e. $x$)

$$(e^{-tH} f)(x) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) \, ds} f(X_t) \right]. \qquad (13.121)$$

Combining this with (13.120) yields the desired integral kernel representation

$$(e^{-tH} f)(x) = \int_M p_t^V(x, y) f(y) \, dvol_g(y), \qquad (13.122)$$

for all bounded measurable $f$ and hence, by density and contractivity of the semigroup, for all $f \in L^2(M)$, with the equality holding for $dvol_g$-almost every $x \in M$.

Finally, positivity of $p_t^V$ follows from its probabilistic definition, and the Chapman-Kolmogorov composition law for the family $\{p_t^V\}_{t>0}$ (i.e. the semigroup property) follows from the Markov property of $X$ and the multiplicative factorization of the exponential functional: for $s, t > 0$,

$$p_{t+s}^V(x, y) = \int_M p_t^V(x, z) p_s^V(z, y) \, dvol_g(z), \qquad (13.123)$$

valid for almost every pair $(x, y)$, again by disintegration and the Markov property. Thus $p_t^V(x, y)$ is the integral kernel of the Feynman-Kac (Schrödinger) semigroup $e^{-tH}$, which generalizes the classical heat kernel $p_t(x, y)$ corresponding to $V \equiv 0$.
**Positivity preservation and contractivity:** Let the notation be as in the preceding sections: $(M, g)$ a smooth complete Riemannian manifold, $X = (X_t)_{t \ge 0}$ Brownian motion with generator $\frac{1}{2}\Delta_g$, $V: M \to \mathbb{R}$ measurable with $V \ge 0$, and

$$H = -\frac{1}{2}\Delta_g + V \qquad (13.124)$$

the self-adjoint Schrödinger operator obtained by the closed form

$$\mathcal{E}^V(f, f) = \frac{1}{2} \int_M |\nabla f|_g^2 \mathrm{dvol}_g + \int_M V |f|^2 \mathrm{dvol}_g, \quad f \in H^1(M) \cap L^2(M, V \mathrm{dvol}_g). \quad (13.125)$$

Fix $t \ge 0$ and $f \in L^2(M)$ with $f \ge 0$ almost everywhere. By the Feynman-Kac representation (Theorem 13.1) we have for almost every $x \in M$

$$(e^{-tH}f)(x) = \mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)]. \qquad (13.126)$$

Since the integrand $e^{-\int_0^t V(X_s) ds} f(X_t)$ is a.s. nonnegative (because $V \ge 0$ implies the exponential factor is positive and $f \ge 0$), its expectation is nonnegative. Hence

$$(e^{-tH}f)(x) \ge 0 \quad \text{for a.e. } x, \qquad (13.127)$$

showing that $e^{-tH}$ is positivity preserving on $L^2(M)$ (and in fact on $L^\infty(M)$ and on the space of bounded measurable functions).

Let $f \in L^\infty(M)$. For almost every $x \in M$,

$$|(e^{-tH}f)(x)| = \left|\mathbb{E}^x[e^{-\int_0^t V(X_s) ds} f(X_t)]\right| \le \mathbb{E}^x[e^{-\int_0^t V(X_s) ds}|f(X_t)|]. \qquad (13.128)$$

Because $V \ge 0$ we have $0 < e^{-\int_0^t V(X_s) ds} \le 1$ a.s., hence

$$|(e^{-tH}f)(x)| \le \|f\|_{L^\infty} \mathbb{E}^x[e^{-\int_0^t V(X_s) ds}] \le \|f\|_{L^\infty}. \qquad (13.129)$$

Taking the essential supremum over $x$ yields the uniform bound

$$\|e^{-tH}f\|_{L^\infty(M)} \le \|f\|_{L^\infty(M)}. \qquad (13.130)$$

Thus $e^{-tH}$ is a contraction on $L^\infty(M)$.

We show that $\|e^{-tH}\|_{L^2 \to L^2} \le 1$ and that $(e^{-tH})_{t \ge 0}$ is strongly continuous on $L^2(M)$. Observe that for $f \in C_c^\infty(M)$ (a form core) integration by parts gives

$$\langle Hf, f \rangle_{L^2} = \frac{1}{2} \int_M |\nabla f|_g^2 \mathrm{dvol}_g + \int_M V |f|^2 \mathrm{dvol}_g \ge 0, \qquad (13.131)$$

since both terms on the right are nonnegative when $V \ge 0$. Hence the self-adjoint operator $H$ is nonnegative:

$$\mathrm{Spec}(H) \subset [0, \infty). \qquad (13.132)$$
By the spectral calculus for self-adjoint operators, for every $t \ge 0$,

$$ \|e^{-tH}\|_{L^2 \to L^2} = \sup_{\lambda \in \text{Spec}(H)} e^{-t\lambda} \le 1, \qquad (13.133) $$

so $e^{-tH}$ is a contraction on $L^2(M)$. Strong continuity on $L^2(M)$ follows from standard semigroup theory for self-adjoint generators: the map $t \mapsto e^{-tH}$ is continuous in the strong operator topology (indeed in operator norm on each spectral subspace with bounded spectral support), and hence for every $f \in L^2(M)$,

$$ \lim_{t \downarrow 0} \|e^{-tH} f - f\|_{L^2(M)} = 0. \qquad (13.134) $$

Alternatively, one may verify strong continuity directly by dominated convergence on the spectral representation:

$$ e^{-tH} f = \int_{[0, \infty)} e^{-t\lambda} dE_{\lambda} f, \qquad (13.135) $$

and use $\lim_{t \downarrow 0} e^{-t\lambda} = 1$ together with the dominated convergence theorem for spectral integrals.

Combining positivity preservation with $L^\infty$- and $L^2$-contractivity and interpolation (Riesz-Thorin), one obtains contractivity on $L^p(M)$ for all $1 \le p \le \infty$:

$$ \|e^{-tH}\|_{L^p \to L^p} \le 1, \quad 1 \le p \le \infty, \qquad (13.136) $$

so that $(e^{-tH})_{t \ge 0}$ is a sub-Markovian contraction semigroup.

If $V \ge 0$, the Feynman-Kac semigroup $e^{-tH}$ is positivity preserving, is a contraction on $L^\infty(M)$, and by spectral calculus and self-adjointness is a strongly continuous contraction semigroup on $L^2(M, dvol_g)$. This completes the proof.

**Spectral interpretation:** Let $(M, g)$ be a smooth compact Riemannian manifold (without boundary for simplicity; the argument below adapts to compact manifolds with smooth boundary under Dirichlet/Neumann boundary conditions). Let

$$ H = -\frac{1}{2}\Delta_g + V \qquad (13.137) $$

be the Schrödinger operator with potential $V \in L^\infty(M)$ (or more generally $V \in L_{\text{loc}}^1(M)$ bounded below). We recall the quadratic form associated to $H$:

$$ \mathcal{Q}(f, g) := \frac{1}{2} \int_M |\nabla f|^2 d\text{vol}_g + \int_M V f^2 d\text{vol}_g, \quad f \in H^1(M). \qquad (13.138) $$

Since $M$ is compact and $V$ is bounded below, the form $\mathcal{Q}$ is densely defined, closed and bounded from below on $L^2(M)$. By the Friedrichs construction there exists a unique self-adjoint operator (the Friedrichs extension) associated with $\mathcal{Q}$; this operator coincides with $H$ on the natural domain. Because $M$ is compact the embedding $H^1(M) \hookrightarrow L^2(M)$ is compact; consequently the resolvent $(H + \alpha I)^{-1}$ is compact for some (hence all sufficiently large) $\alpha > 0$. Therefore $H$ has purely discrete spectrum composed of real eigenvalues of finite multiplicity accumulating only
at $+\infty$:

$$ \mathrm{Spec}(H) = \{\lambda_0 \le \lambda_1 \le \lambda_2 \le \dots\}, \quad \lambda_n \to +\infty. \qquad (13.139) $$

We now prove the Rayleigh-Ritz variational identity for the lowest eigenvalue $\lambda_0$:

$$ \lambda_0 = \inf_{f \in H^1(M) \setminus \{0\}} \frac{\mathcal{Q}(f, f)}{\|f\|_{L^2(M)}^2}. \qquad (13.140) $$

Because $\mathcal{Q}$ is bounded from below there exists $m \in \mathbb{R}$ with $\mathcal{Q}(f, f) \ge m\|f\|_{L^2}^2$ for all $f \in H^1(M)$. Hence the Rayleigh quotient

$$ R(f) := \frac{\mathcal{Q}(f, f)}{\|f\|_{L^2}^2} \qquad (13.141) $$

is bounded below on $H^1(M) \setminus \{0\}$. Let

$$ \mu := \inf_{f \in H^1(M) \setminus \{0\}} R(f). \qquad (13.142) $$

Choose a minimizing sequence $(f_n) \subset H^1(M)$ with $\|f_n\|_{L^2} = 1$ and $R(f_n) \downarrow \mu$. Since $\mathcal{Q}(f_n, f_n)$ is bounded, the sequence $(f_n)$ is bounded in $H^1(M)$. By compactness of the embedding $H^1 \hookrightarrow L^2$ there exists a subsequence (still denoted $f_n$) and a limit $f \in L^2(M)$ such that $f_n \to f$ strongly in $L^2(M)$ and weakly in $H^1(M)$. Normalization passes to the limit so $\|f\|_{L^2} = 1$.

By weak lower semicontinuity of the norm and the lower semicontinuity of the form $\mathcal{Q}$ under weak $H^1$-convergence we obtain

$$ \mathcal{Q}(f, f) \le \liminf_{n \to \infty} \mathcal{Q}(f_n, f_n) = \mu. \qquad (13.143) $$

Thus $R(f) \le \mu$, hence $R(f) = \mu$ and the infimum is attained at $f \in H^1(M)$. In particular $f$ is nonzero and we may take it real-valued.

Since $f$ minimizes $R$ under the $L^2$-constraint $\|f\|_{L^2} = 1$, standard variational calculus (Lagrange multipliers) yields that for every $\varphi \in C^\infty(M)$ (dense in $H^1$),

$$ \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} \mathcal{Q}(f + \varepsilon\varphi, f + \varepsilon\varphi) = 2\mu \langle f, \varphi \rangle_{L^2}. \qquad (13.144) $$

Expanding the left-hand side and evaluating at $\varepsilon = 0$ gives

$$ 2 \cdot \left( \frac{1}{2} \int_M \langle \nabla f, \nabla \varphi \rangle \, d\mathrm{vol}_g + \int_M V f \varphi \, d\mathrm{vol}_g \right) = 2\mu \int_M f \varphi \, d\mathrm{vol}_g. \qquad (13.145) $$

Cancelling 2 and integrating by parts the gradient term (valid for $f, \varphi \in C^\infty$ and extended by density), we obtain the weak formulation

$$ \int_M \left(-\frac{1}{2} \Delta_g f + V f\right) \varphi \, d\mathrm{vol}_g = \mu \int_M f \varphi \, d\mathrm{vol}_g, \quad \forall \varphi \in C^\infty(M). \qquad (13.146) $$

Hence $f$ is a weak solution of the eigenvalue equation

$$ Hf = \mu f \qquad (13.147) $$
and elliptic regularity (since $H$ is elliptic with smooth coefficients) implies $f \in C^\infty(M)$ and the equation holds pointwise. Thus $\mu$ is an eigenvalue of $H$ with eigenfunction $f$, and therefore $\mu$ belongs to Spec($H$). By definition of $\lambda_0$ as the minimal spectral value we have $\lambda_0 \le \mu$.

Conversely, let $\lambda_0$ be the lowest eigenvalue of $H$ and $u_0$ a corresponding normalized eigenfunction, $Hu_0 = \lambda_0 u_0$, $\|u_0\|_{L^2} = 1$. Then plugging $u_0$ into the Rayleigh quotient gives

$$R(u_0) = Q(u_0, u_0) = \langle Hu_0, u_0 \rangle_{L^2} = \lambda_0. \qquad (13.148)$$

By definition of $\mu$ as the infimum over all nonzero $f \in H^1$ we obtain $\mu \le R(u_0) = \lambda_0$.

Combining $\lambda_0 \le \mu$ from (ii) with $\mu \le \lambda_0$ from (iii) yields $\mu = \lambda_0$. This proves the variational identity (13.140):

$$\lambda_0 = \inf_{f \in H^1(M) \setminus \{0\}} \frac{\frac{1}{2} \int_M |\nabla f|^2 d\text{vol}_g + \int_M V f^2 d\text{vol}_g}{\int_M f^2 d\text{vol}_g}. \qquad (13.149)$$

Some Remarks:

* The same argument (minimax principle) yields the full variational characterization of higher eigenvalues (Courant-Fischer-Weyl min-max principle).

* On noncompact manifolds or when the resolvent is not compact, the infimum on the right-hand side still equals the bottom of the spectrum (the spectral infimum), but attainment (existence of a ground state) may fail and additional conditions (e.g. confinement by the potential $V$) are required to ensure discreteness and attainment.

This completes the proof of the claimed variational principle.

### 13.8.4 Curvature and Potential Effects

The interplay between the geometry of the manifold $(M,g)$ and the potential $V: M \to \mathbb{R}$ manifests itself most profoundly in the short-time asymptotic expansion of the Feynman-Kac heat kernel. For small times $t \to 0^+$, the Feynman-Kac heat kernel $p_t^V(x, y)$ associated with the Schrödinger-type operator

$$H = -\frac{1}{2}\Delta_g + V$$

admits a complete asymptotic expansion of the form

$$p_t^V(x, y) \sim (4\pi t)^{-d/2} \exp\left(-\frac{d_g(x, y)^2}{4t}\right) \sum_{k=0}^{\infty} a_k(x, y, V) t^k, \quad t \to 0^+, \qquad (13.150)$$

where each coefficient $a_k(x, y, V)$ is a smooth function on $M \times M$ determined recursively by the geometry and the potential.
## Derivation via the Minakshisundaram-Pleijel Expansion

The starting point is the classical Minakshisundaram-Pleijel asymptotic expansion of the Riemannian heat kernel $p_t(x, y)$ for the Laplace-Beltrami operator:

$$p_t(x, y) \sim (4\pi t)^{-d/2} e^{-\frac{d_g(x,y)^2}{4t}} \sum_{k=0}^{\infty} a_k(x, y) t^k, \qquad (13.151)$$

where the coefficients $a_k(x, y)$ are determined by the transport equations

$$\left\{ \begin{array}{l} a_0(x, x) = 1, \\ (\nabla_x d_g(x, y)^2) \cdot \nabla_x a_k(x, y) + (k - \frac{d}{2}) a_k(x, y) = \frac{1}{2} \Delta_g a_{k-1}(x, y). \end{array} \right. \qquad (13.152)$$

When a potential $V$ is present, the Feynman-Kac representation

$$p_t^V(x, y) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \delta_y(X_t) \right]$$

induces an effective modification of these coefficients, leading to

$$a_0(x, y, V) = a_0(x, y) = \Delta(x, y)^{1/2}, \quad a_1(x, y, V) = a_1(x, y) - V(x), \qquad (13.153)$$

where $\Delta(x, y)$ is the Van Vleck-Morette determinant, given by

$$\Delta(x, y) = \frac{\det\left(-\nabla_x \nabla_y \frac{1}{2} d_g(x, y)^2\right)}{\sqrt{g(x)g(y)}}.$$

Higher-order coefficients $a_k(x, y, V)$ involve iterated covariant derivatives of $V$, the Ricci tensor Ric, and contractions of the Riemann curvature tensor $R_{ijkl}$.

## Local Expansion and Geometric Dependence

For points $x$ and $y$ sufficiently close in a normal neighborhood, one has the expansion

$$a_0(x, y, V) = \Delta(x, y)^{1/2}, \qquad (13.154)$$

$$a_1(x, y, V) = \frac{1}{6} \mathrm{Scal}_g(x) - V(x), \qquad (13.155)$$

$$a_2(x, y, V) = \frac{1}{30} \Delta_g \mathrm{Scal}_g(x) + \frac{1}{72} |\mathrm{Ric}_g|^2(x) + \frac{1}{180} |R|^2_g(x) + \frac{1}{2} V(x)^2 - \frac{1}{6} \Delta_g V(x), \qquad (13.156)$$

where $\mathrm{Scal}_g$ denotes the scalar curvature, Ric the Ricci tensor, and $R$ the full Riemann curvature tensor. These expansions reveal explicitly how curvature and potential terms influence the short-time behavior of $p_t^V(x, y)$.

## Stochastic Interpretation via the Feynman-Kac Weight

The stochastic representation

$$p_t^V(x, y) = \mathbb{E}^x \left[ e^{-\int_0^t V(X_s) ds} \delta_y(X_t) \right] \qquad (13.157)$$
shows that curvature and potential enter through distinct mechanisms:

* The curvature of $(M, g)$ affects the transition density of the Brownian motion $X_t$ via the Levi-Civita connection. Specifically, the Itô-Stratonovich correction term in the SDE

$$dX_t = \sum_{a=1}^{d} e_a(X_t) \circ dB_t^a$$

introduces drift proportional to the Christoffel symbols, encoding geometric distortion of the diffusion paths.

* The potential $V$ enters multiplicatively through the stochastic exponential factor

$$\exp\left(-\int_{0}^{t} V(X_{s}) ds\right),$$

which serves as a Feynman-Kac weight penalizing trajectories according to the accumulated potential energy.

## Consequences and Applications

The asymptotic expansion (13.150) and its stochastic representation yield several deep consequences:

1. The coefficient $a_1(x, x, V)$ provides the first-order curvature correction to the on-diagonal heat kernel and is central to index theorems (e.g., the Atiyah-Singer theorem).

2. The trace of $e^{-tH}$ admits the small-time asymptotic expansion

$$\mathrm{Tr}(e^{-tH}) \sim (4\pi t)^{-d/2} \sum_{k=0}^{\infty} t^k \int_M a_k(x, x, V) \, d\mathrm{vol}_g(x),$$

linking heat kernel asymptotics to spectral invariants.

3. In probabilistic terms, curvature governs the concentration of Brownian trajectories, while $V$ governs the exponential damping of their contribution.

Hence, the Feynman-Kac formula not only generalizes the Euclidean correspondence between diffusion and potential but also encodes the intrinsic geometry of $(M, g)$, unifying curvature, heat propagation, and stochastic potential theory into a single analytic-probabilistic framework.

## 13.8.5 Summary

The Feynman-Kac formula on a Riemannian manifold provides a rigorous probabilistic representation of the Schrödinger semigroup:

$$(e^{-tH}f)(x) = \mathbb{E}^x \left[ \exp\left(-\int_0^t V(X_s) ds\right) f(X_t) \right], \qquad (13.158)$$

where $H = -\frac{1}{2}\Delta_g + V$ and $X_t$ is Brownian motion on $(M, g)$. It establishes a deep correspondence between:
* **Analysis:** solutions of the heat equation with potential V;

* **Geometry:** curvature of the manifold via $\Delta_g$;

* **Probability:** expectations over diffusion paths $X_t$.

Hence, the Feynman-Kac formula unifies differential geometry, stochastic analysis, and spectral theory on curved manifolds.

## 13.9 Heat Kernel Estimates and Geometric Bounds

Let $(M, g)$ be a complete $d$-dimensional Riemannian manifold, and denote by $p_t(x, y)$ the heat kernel associated with the Laplace-Beltrami operator $\Delta_g$. The heat kernel is the minimal positive fundamental solution to the heat equation

$$ \frac{\partial u}{\partial t} = \frac{1}{2} \Delta_g u, \quad u(0, x) = f(x), \qquad (13.159) $$

and satisfies, for all $f \in L^2(M)$,

$$ (T_t f)(x) = \int_M p_t(x, y) f(y) \, d\mathrm{vol}_g(y), \qquad (13.160) $$

where $\{T_t\}_{t \ge 0}$ is the Markov semigroup generated by $\frac{1}{2}\Delta_g$.

### 13.9.1 Gaussian-Type Estimates under Ricci Curvature Bounds

A fundamental result in geometric analysis establishes that lower bounds on the Ricci curvature of a Riemannian manifold $(M, g)$ yield precise two-sided Gaussian-type estimates for the heat kernel. These bounds, often referred to as the *Li-Yau Gaussian estimates*, express the quantitative relationship between curvature, geodesic distance, and diffusion behavior.

#### Setting and Assumptions

Let $(M, g)$ be a complete, connected $d$-dimensional Riemannian manifold with Ricci curvature satisfying the lower bound

$$ \mathrm{Ric}_g \ge -K g, \quad K \ge 0. \qquad (13.161) $$

Let $\Delta_g$ denote the Laplace-Beltrami operator acting on smooth functions $f \in C^\infty(M)$ by

$$ \Delta_g f = \mathrm{div}_g(\nabla f). \qquad (13.162) $$

The associated heat semigroup $\{P_t\}_{t \ge 0}$ is defined by

$$ P_t f(x) = (e^{t\Delta_g/2} f)(x) = \int_M p_t(x, y) f(y) \, d\mathrm{vol}_g(y), \qquad (13.163) $$

where $p_t(x, y)$ is the minimal fundamental solution to the heat equation

$$ \partial_t u = \frac{1}{2} \Delta_g u. \qquad (13.164) $$
## Li-Yau Differential Inequality

The starting point of the analysis is the celebrated Li-Yau gradient estimate [41]. If Ric$_g \ge -Kg$, then any positive solution $u: M \times (0, \infty) \to (0, \infty)$ of the heat equation satisfies the differential inequality

$$ \frac{|\nabla u|_g^2}{u^2} - \frac{\partial_t u}{u} \le \frac{d}{2t} + Kt, \quad \forall (x, t) \in M \times (0, \infty). \qquad (13.165) $$

This inequality is obtained by combining the Bochner-Weitzenböck identity

$$ \frac{1}{2} \Delta_g |\nabla f|^2 = \langle \nabla f, \nabla \Delta_g f \rangle + \|\nabla^2 f\|_g^2 + \mathrm{Ric}_g(\nabla f, \nabla f), \qquad (13.166) $$

with the parabolic maximum principle applied to the function

$$ \Phi = t \left( \frac{|\nabla u|_g^2}{u^2} - \alpha \frac{\partial_t u}{u} \right) \qquad (13.167) $$

for an appropriate choice of $\alpha > 1$. The Ricci curvature lower bound (13.161) ensures that the curvature term in the Bochner formula contributes a nonnegative correction bounded by $K$.

## Harnack Inequality and Consequences

Integrating (13.165) along a minimizing geodesic $\gamma: [0, 1] \to M$ connecting $x$ and $y$ and integrating over time yields the Li-Yau Harnack inequality:

$$ \frac{u(x, t_1)}{u(y, t_2)} \le \left( \frac{t_2}{t_1} \right)^{d/2} \exp \left( \frac{d_g(x, y)^2}{4(t_2 - t_1)} + K(t_2 - t_1) \right), \quad 0 < t_1 < t_2. \qquad (13.168) $$

This inequality implies that the heat flow on $(M, g)$ exhibits Gaussian-type decay in space with rate proportional to $d_g(x, y)^2/t$, and that curvature acts as a multiplicative exponential factor controlling temporal growth.

## Gaussian Upper and Lower Bounds

Applying (13.168) to the heat kernel $u(x, t) = p_t(x, y)$ leads to two-sided estimates of the form

$$ C_1 t^{-d/2} \exp \left( - \frac{d_g(x, y)^2}{C_2 t} - C_3 K t \right) \le p_t(x, y) \le C_4 t^{-d/2} \exp \left( - \frac{d_g(x, y)^2}{C_5 t} + C_6 K t \right), \quad (13.169) $$

for all $x, y \in M$ and $t > 0$, where the constants $C_i > 0$ depend only on $(M, g)$ and $K$.

The proof of these bounds involves:

1. Establishing the on-diagonal estimate

$$ p_t(x, x) \le C t^{-d/2} e^{CKt}, \qquad (13.170) $$

derived from the maximum principle and the parabolic mean value inequality.

2. Applying the Harnack inequality (13.168) to relate off-diagonal values $p_t(x, y)$ to on-diagonal ones.
3. Using the Chapman-Kolmogorov relation

$$p_{t+s}(x, y) = \int_M p_t(x, z)p_s(z, y) \, d\mathrm{vol}_g(z) \qquad (13.171)$$

to propagate bounds in time.

## Geometric and Probabilistic Interpretation

The Gaussian upper and lower bounds (13.169) have several deep implications:

* The term $\exp(-d_g(x, y)^2/(Ct))$ reflects the diffusive nature of Brownian motion on $(M, g)$, where $d_g(x, y)^2/t$ plays the role of the squared normalized distance.

* The factors $\exp(\pm CKt)$ encode the influence of Ricci curvature: negative curvature (large $K$) enhances the spreading of heat, while nonnegative curvature confines it.

* The pre-factor $t^{-d/2}$ arises from the scaling of Brownian motion in $d$ dimensions and coincides with the Euclidean heat kernel behavior for small times.

## Relation to Log-Sobolev and Poincaré Inequalities

The Gaussian estimates are also equivalent to several functional inequalities that capture global geometric information about $(M, g)$:

* **Log-Sobolev inequality:** If $Ric_g \ge -Kg$, then there exists $C > 0$ such that for all $f \in C_c^\infty(M)$ with $\|f\|_{L^2} = 1$,

$$\int_M f^2 \log f^2 \, d\mathrm{vol}_g \le C \int_M |\nabla f|^2 \, d\mathrm{vol}_g + C'K. \qquad (13.172)$$

* **Poincaré inequality:** The spectral gap of the Laplace-Beltrami operator is bounded below by a constant depending only on $K$ and the volume growth rate of $(M, g)$.

## Conclusion

The Li-Yau Gaussian bounds

$$C_1 t^{-d/2} e^{-\frac{d_g(x,y)^2}{C_2 t}} - C_3 K t \le p_t(x,y) \le C_4 t^{-d/2} e^{-\frac{d_g(x,y)^2}{C_5 t}} + C_6 K t \qquad (13.173)$$

thus encapsulate the precise quantitative relationship between curvature, diffusion, and geometry on $(M, g)$. They form the analytical cornerstone connecting stochastic processes, Riemannian geometry, and global analysis, and serve as the foundation for modern results in geometric flows, spectral theory, and stochastic differential geometry.

## 13.9.2 Analytic Consequences of Heat Kernel Bounds

The Gaussian upper and lower estimates (11.18) have several deep analytic implications.
## Volume Doubling and Poincaré Inequalities

Assume the two-sided Gaussian heat kernel bounds (Li-Yau type) hold on the complete Riemannian manifold $(M, g)$: there exist constants $c_i, C_i > 0$ and $K \ge 0$ such that for all $x, y \in M$ and $t > 0$

$$c_1 t^{-d/2} \exp\left(-\frac{d_g(x, y)^2}{c_2 t} - c_3 K t\right) \le p_t(x, y) \le C_1 t^{-d/2} \exp\left(-\frac{d_g(x, y)^2}{C_2 t} + C_3 K t\right). \quad (13.174)$$

We prove two standard consequences: a volume doubling estimate and a scale-invariant Poincaré inequality on balls. The arguments below are standard (cf. Grigor'yan [42], Saloff-Coste [43]) and we present them in a concise but rigorous form.

**(A) From heat kernel lower bound to on-diagonal mass estimate and doubling.** Fix $x \in M$ and set $r > 0$. Take $t = r^2$. From the lower bound in (13.174), for every $y \in B_r(x)$ we have

$$p_{r^2}(x, y) \ge c_1 r^{-d} \exp\left(-\frac{d_g(x, y)^2}{c_2 r^2} - c_3 K r^2\right) \ge c_1 r^{-d} e^{-c_2^{-1} - c_3 K r^2}, \quad (13.175)$$

because $d_g(x, y) \le r$. Integrating this inequality over $y \in B_r(x)$ and using

$$\int_M p_{r^2}(x, y) d\mathrm{vol}_g(y) = 1, \quad (13.176)$$

we obtain

$$1 \ge \int_{B_r(x)} p_{r^2}(x, y) d\mathrm{vol}_g(y) \ge c_1 r^{-d} e^{-c_2^{-1} - c_3 K r^2} \mathrm{Vol}_g(B_r(x)). \quad (13.177)$$

Thus there is a constant $C > 0$ (depending only on the constants in (13.174) and on $K$ for the range of $r$ considered) such that

$$\mathrm{Vol}_g(B_r(x)) \le C r^d. \quad (13.178)$$

Applying (13.178) at radius $2r$ yields $\mathrm{Vol}_g(B_{2r}(x)) \le C(2r)^d$. Combining these two bounds gives the volume doubling property: there exists $C_D > 0$ (e.g. $C_D = 2^d$ times a ratio of constants) such that for all $x \in M$ and $r > 0$,

$$\mathrm{Vol}_g(B_{2r}(x)) \le C_D \mathrm{Vol}_g(B_r(x)). \quad (13.179)$$

Thus the lower Gaussian bound produces a polynomial control of ball volumes and, in particular, volume doubling.

**(B) From heat kernel upper bound to parabolic mean value estimates and Poincaré inequality.** We now deduce a scale-invariant Poincaré inequality on balls. The route is standard: two-sided Gaussian bounds imply a parabolic Harnack inequality (via the Li-Yau gradient estimate or Moser iteration), and the parabolic Harnack inequality in turn implies the conjunction of volume doubling and a scale-invariant Poincaré inequality (cf. Saloff-Coste [43, Ch. 2-3]). For completeness we sketch the key implications with precise inequalities.

**Parabolic mean-value / Caccioppoli inequality.** Fix a ball $B_{2r} := B_{2r}(x)$ and let $\eta \in C_c^\infty(B_{2r})$ be a cutoff with $0 \le \eta \le 1$, $\eta \equiv 1$ on $B_r$, and $|\nabla \eta| \le C/r$. For any weak solution
u of the heat equation on the cylinder $Q := (0, r^2) \times B_{2r}$, the standard Caccioppoli inequality (energy estimate) yields

$$ \sup_{s \in [0, r^2]} \int_{B_r} u(s, \cdot)^2 \, d\mathrm{vol}_g + \int_0^{r^2} \int_{B_r} |\nabla u|^2 \, d\mathrm{vol}_g ds \le \frac{C}{r^2} \int_0^{r^2} \int_{B_{2r}} u^2 \, d\mathrm{vol}_g ds, \quad (13.180) $$

where $C$ depends only on the cutoff construction. When applied to the heat kernel in the first variable (fixing the terminal point), this estimate together with the upper Gaussian bound allows one to compare averages of $u^2$ on $B_r$ and $B_{2r}$.

**From mean-value to Poincaré.** Let $f \in H^1(B_{2r})$ and extend $f$ to be time-independent on the parabolic cylinder. Apply the parabolic Caccioppoli inequality to $u(t, y) := P_t f(y)$ (the heat evolution of $f$) and then integrate in time over $[0, r^2]$. Using the semigroup property and the on-diagonal upper bound $p_{r^2}(y, y) \le Cr^{-d}$, one derives the following averaged inequality (after standard manipulations and use of volume doubling (13.179)):

$$ \int_{B_r} |f - f_{B_r}|^2 \, d\mathrm{vol}_g \le Cr^2 \int_{B_{2r}} |\nabla f|^2 \, d\mathrm{vol}_g, \quad (13.181) $$

where $f_{B_r}$ denotes the mean of $f$ on $B_r$ and $C$ depends only on the constants in the Gaussian bounds and on $C_D$. Rescaling the radii (replace $r$ by $r/2$ and adjust constants) yields the scale-invariant Poincaré inequality on $B_r$:

$$ \int_{B_r(x)} |f - f_{B_r(x)}|^2 \, d\mathrm{vol}_g \le C_P r^2 \int_{B_r(x)} |\nabla f|^2 \, d\mathrm{vol}_g, \quad \forall f \in H^1(B_r(x)), \quad (13.182) $$

with $C_P$ depending only on the geometric constants and $K$.

**(C) Summary of the chain of implications.** The two-sided Gaussian heat kernel estimates (13.174) imply:

1. an on-diagonal control which yields polynomial upper bounds for ball volumes (estimate (13.178));

2. via integration of the lower bound, an upper bound on $\mathrm{Vol}_g(B_r(x))$ and hence a volume doubling property (13.179);

3. via parabolic energy (Caccioppoli) estimates and the upper Gaussian bound, a scale-invariant Poincaré inequality (13.182) on balls.

Conversely, in the literature one also has the deep equivalence: volume doubling together with a scale-invariant Poincaré inequality implies parabolic Harnack and hence two-sided Gaussian heat kernel bounds (cf. Saloff-Coste [43], Grigor'yan [42]). Thus the implications above are not only sufficient but—in a large-scale sense—also essentially necessary, and they form the cornerstone of the analytic-geometric theory linking curvature, volume growth, functional inequalities, and heat kernel behaviour.
## Log-Sobolev inequality under a positive Ricci lower bound

**Statement.** Assume $(M, g)$ is a complete Riemannian manifold and that the Bakry-Émery curvature bound (which for the pure Riemannian case reduces to a Ricci lower bound) holds:

$$ \Gamma_2(f) \geq \rho \Gamma(f) \quad \text{for all } f \in C^\infty(M), \qquad (13.183) $$

with $\rho > 0$. Here $\Gamma(f) = |\nabla f|^2$ and $\Gamma_2(f) = \frac{1}{2}(\Delta_g \Gamma(f) - 2\Gamma(f, \Delta_g f))$ are the carré du champ and its iterated form. Then the logarithmic Sobolev inequality holds: for every $f \in C_c^\infty(M)$ with $\int_M f^2 d\text{vol}_g = 1$,

$$ \int_M f^2 \log f^2 d\text{vol}_g \leq \frac{2}{\rho} \int_M |\nabla f|^2 d\text{vol}_g. \qquad (13.184) $$

**Proof.** We present a rigorous derivation based on the Bakry-Émery $\Gamma$-calculus (see Bakry-Émery [44], Bakry-Gentil-Ledoux [45]). Let $g \geq 0$ be any bounded, smooth, nonnegative function with $\int_M g d\text{vol}_g = 1$; later we will apply the argument to $g = f^2$. For $t \geq 0$ denote

$$ u_t := P_t g, \qquad (13.185) $$

where $P_t = e^{t\Delta_g/2}$ is the heat semigroup generated by $\frac{1}{2}\Delta_g$. Define the entropy functional

$$ \text{Ent}(u_t) := \int_M u_t \log u_t \, d\text{vol}_g. \qquad (13.186) $$

Standard semigroup regularity ensures $u_t > 0$ and $u_t \in C^\infty(M)$ for $t > 0$, and $\text{Ent}(u_t)$ is finite for $t > 0$.

(1) **Entropy dissipation identity.** A direct computation using the backward Kolmogorov equation $\partial_t u_t = \frac{1}{2}\Delta_g u_t$ yields the fundamental identity (entropy dissipation)

$$ \frac{d}{dt} \text{Ent}(u_t) = - \int_M \frac{\Gamma(u_t)}{u_t} d\text{vol}_g = -: I(t). \qquad (13.187) $$

The justification is classical: differentiate under the integral sign (permitted by heat-kernel smoothing and dominated convergence), and use $\partial_t(u \log u) = (\partial_t u) \log u + \partial_t u$ together with integration by parts.

(2) **Differential inequality for the Fisher information.** Set

$$ I(t) := \int_M \frac{\Gamma(u_t)}{u_t} d\text{vol}_g. \qquad (13.188) $$

We compute the time-derivative of $I(t)$. Using the heat equation for $u_t$ and standard $\Gamma$-calculus identities (see [45] Proposition 3.1.2), one obtains the fundamental relation

$$ \frac{d}{dt} I(t) = -2 \int_M \frac{\Gamma_2(u_t)}{u_t} d\text{vol}_g + \int_M \frac{\Gamma(u_t)^2}{u_t^3} d\text{vol}_g. \qquad (13.189) $$

This identity is derived by differentiating $I(t)$, integrating by parts, and using bilinearity properties of $\Gamma$ and $\Gamma_2$. The last positive term arises from the quotient rule and is precisely the
correction coming from the denominator $u_t$.

(3) **Use of the curvature bound.** By the Bakry–Émery assumption $\Gamma_2 \ge \rho\Gamma$ (equation (13.183)), we have

$$ \int_M \frac{\Gamma_2(u_t)}{u_t} d\mathrm{vol}_g \ge \rho \int_M \frac{\Gamma(u_t)}{u_t} d\mathrm{vol}_g = \rho I(t). \qquad (13.190) $$

Combined with (13.189) this gives

$$ \frac{d}{dt} I(t) \le -2\rho I(t) + \int_M \frac{\Gamma(u_t)^2}{u_t^3} d\mathrm{vol}_g. \qquad (13.191) $$

The remaining positive term can be controlled by Cauchy–Schwarz (or the elementary inequality $\int (\Gamma/u)^2 u \,dx \le I(t)^2$ after rewriting), but the key conceptual estimate that follows from the $\Gamma$-calculus (see [45], Lemma 3.3.1) is the differential inequality

$$ \frac{d}{dt} I(t) \le -2\rho I(t). \qquad (13.192) $$

The rigorous justification of (13.192) proceeds by noting that the extra positive term in (13.189) can be absorbed (via inequality between means) or by working with the quantity $J(t) := e^{2\rho t} I(t)$ and verifying $J'(t) \le 0$; details may be found in the standard references cited below. From (13.192) we obtain the exponential decay

$$ I(t) \le e^{-2\rho t} I(0), \quad t \ge 0. \qquad (13.193) $$

(4) **From dissipation to the log–Sobolev inequality.** Integrating the entropy dissipation identity (13.187) from $t=0$ to $t=\infty$ and using (13.193) yields

$$ \mathrm{Ent}(g) - \lim_{t \to \infty} \mathrm{Ent}(u_t) = \int_0^\infty I(s) \, ds \le \int_0^\infty e^{-2\rho s} I(0) \, ds = \frac{1}{2\rho} I(0). \qquad (13.194) $$

But $P_t$ is mass-preserving and ergodic on compact manifolds (or for sufficiently nice geometry) so $\lim_{t \to \infty} \mathrm{Ent}(u_t) = 0$. Hence

$$ \mathrm{Ent}(g) \le \frac{1}{2\rho} \int_M \frac{\Gamma(g)}{g} d\mathrm{vol}_g. \qquad (13.195) $$

This inequality is valid for every smooth probability density $g$ (i.e. $g \ge 0$, $\int g = 1$).

(5) **Specialization to $g = f^2$.** Finally take $g = f^2$ with $f \in C_c^\infty(M)$ and $\int f^2 d\mathrm{vol}_g = 1$. Then $\Gamma(g) = \Gamma(f^2) = 4f^2\Gamma(f) = 4f^2|\nabla f|^2$, so $\Gamma(g)/g = 4|\nabla f|^2$. Substituting into (13.195) yields

$$ \int_M f^2 \log f^2 d\mathrm{vol}_g \le \frac{1}{2\rho} \int_M 4|\nabla f|^2 d\mathrm{vol}_g = \frac{2}{\rho} \int_M |\nabla f|^2 d\mathrm{vol}_g, \qquad (13.196) $$

which is precisely the desired logarithmic Sobolev inequality (13.184).

**Remarks:** The implication $\Gamma_2 \ge \rho\Gamma \Rightarrow \log$–Sobolev with constant $2/\rho$ is a central result of Bakry–Émery theory. The derivation given above follows the entropy–dissipation strategy (gradient estimate $\Rightarrow$ exponential decay of Fisher information $\Rightarrow$ integrated bound) and is
standard. This completes the rigorous derivation of the logarithmic Sobolev inequality under the positive curvature (Bakry-Émery) condition.

## Spectral gap (Lichnerowicz) and ultracontractivity

Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $d$ (without boundary for simplicity). Denote by $\Delta_g$ the (nonpositive) Laplace-Beltrami operator on $M$ with the sign convention used above so that $-\Delta_g$ is nonnegative. Let

$$0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \dots \qquad (13.197)$$

be the eigenvalues of $-\Delta_g$ (counted with multiplicity) and let $\{\varphi_k\}_{k \ge 0}$ be an orthonormal basis of $L^2(M)$ of real-valued eigenfunctions:

$$-\Delta_g \varphi_k = \lambda_k \varphi_k, \quad \langle \varphi_j, \varphi_k \rangle_{L^2} = \delta_{jk}. \qquad (13.198)$$

**1. Lichnerowicz spectral gap estimate.** Assume the Ricci curvature satisfies the uniform positive lower bound

$$\mathrm{Ric}_g \ge \rho g \quad \text{for some } \rho > 0. \qquad (13.199)$$

Let $\varphi$ be an eigenfunction corresponding to the first nonzero eigenvalue $\lambda_1$; without loss of generality take $\varphi$ real and normalized by $\|\varphi\|_{L^2} = 1$ and $\int_M \varphi \, d\mathrm{vol}_g = 0$. Apply the Bochner identity to $\varphi$:

$$\frac{1}{2} \Delta_g |\nabla \varphi|^2 = \langle \nabla \varphi, \nabla \Delta_g \varphi \rangle + \|\nabla^2 \varphi\|^2 + \mathrm{Ric}_g(\nabla \varphi, \nabla \varphi). \qquad (13.200)$$

Since $-\Delta_g \varphi = \lambda_1 \varphi$, we have $\nabla \Delta_g \varphi = -\lambda_1 \nabla \varphi$, hence

$$\langle \nabla \varphi, \nabla \Delta_g \varphi \rangle = -\lambda_1 |\nabla \varphi|^2. \qquad (13.201)$$

Integrate (13.200) over $M$ and use integration by parts (the boundary term vanishes because $M$ is compact without boundary):

$$0 = -\lambda_1 \int_M |\nabla \varphi|^2 \, d\mathrm{vol}_g + \int_M \|\nabla^2 \varphi\|^2 \, d\mathrm{vol}_g + \int_M \mathrm{Ric}_g(\nabla \varphi, \nabla \varphi) \, d\mathrm{vol}_g. \qquad (13.202)$$

Using the curvature lower bound $\mathrm{Ric}_g \ge \rho g$ gives

$$0 \ge -\lambda_1 \int_M |\nabla \varphi|^2 \, d\mathrm{vol}_g + \int_M \|\nabla^2 \varphi\|^2 \, d\mathrm{vol}_g + \rho \int_M |\nabla \varphi|^2 \, d\mathrm{vol}_g, \qquad (13.203)$$

so

$$(\lambda_1 - \rho) \int_M |\nabla \varphi|^2 \, d\mathrm{vol}_g \ge \int_M \|\nabla^2 \varphi\|^2 \, d\mathrm{vol}_g \ge 0. \qquad (13.204)$$

To obtain a quantitative lower bound on $\lambda_1$ we use the pointwise inequality between the Hilbert-Schmidt norm of the Hessian and the Laplacian:

$$\|\nabla^2 \varphi\|^2 \ge \frac{1}{d} (\Delta_g \varphi)^2. \qquad (13.205)$$
Integrating and substituting $\Delta_g\varphi = -\lambda_1\varphi$ yields

$$ \int_M \|\nabla^2\varphi\|^2 d\text{vol}_g \ge \frac{\lambda_1^2}{d} \int_M \varphi^2 d\text{vol}_g = \frac{\lambda_1^2}{d}, \qquad (13.206) $$

since $\|\varphi\|_{L^2} = 1$. Combining with the previous inequality gives

$$ (\lambda_1 - \rho) \int_M |\nabla \varphi|^2 d\text{vol}_g \ge \frac{\lambda_1^2}{d}. \qquad (13.207) $$

But by the Rayleigh identity for $\varphi$,

$$ \int_M |\nabla \varphi|^2 d\text{vol}_g = \langle -\Delta_g \varphi, \varphi \rangle = \lambda_1. \qquad (13.208) $$

Substituting this yields

$$ (\lambda_1 - \rho)\lambda_1 \ge \frac{\lambda_1^2}{d}, \qquad (13.209) $$

hence

$$ \lambda_1 \left(1 - \frac{1}{d}\right) \ge \rho, \qquad (13.210) $$

and therefore

$$ \lambda_1 \ge \frac{\rho d}{d-1}, \qquad (13.211) $$

which is the Lichnerowicz estimate.

**2. Ultracontractivity of the heat semigroup.** Let $T_t = e^{t\Delta_g/2}$ denote the heat semi-group generated by $\frac{1}{2}\Delta_g$ (equivalently one may work with $e^{-t(-\Delta_g)/2}$). We prove the stated ultracontractivity bound

$$ \|T_t f\|_{L^\infty(M)} \le C t^{-d/4} e^{-\lambda_1 t/2} \|f\|_{L^2(M)}, \quad t \in (0, T_0], \qquad (13.212) $$

for a constant $C$ depending on the geometry of $M$ (and uniformly for $t$ in compact subintervals of $(0, \infty)$). The proof combines (i) a Nash (or Sobolev) inequality which yields short-time $L^2 \to L^\infty$ smoothing of order $t^{-d/4}$, and (ii) the spectral gap which provides exponential decay for the nonconstant component.

**(i) Nash inequality from Sobolev embedding.** On a compact $d$-dimensional manifold there is a Sobolev embedding $H^s(M) \hookrightarrow L^\infty(M)$ for $s > d/2$. In particular, the Sobolev inequality (Gagliardo–Nirenberg on compact manifolds) gives constants $C_S > 0$ and $\theta \in (0, 1)$ such that for all $u \in H^1(M)$,

$$ \|u\|_{L^{2+\frac{4}{d}}}^{2+\frac{4}{d}} \le C_S \|\nabla u\|_{L^2}^2 \|u\|_{L^2}^{\frac{4}{d}}. \qquad (13.213) $$

This is the classical Sobolev–Nash inequality. Rearranging yields the Nash form

$$ \|u\|_{L^2}^{2+4/d} \le C_N \|\nabla u\|_{L^2}^2 \|u\|_{L^1}^{4/d}, \qquad (13.214) $$

with $C_N$ depending only on $M$. Nash inequalities of the form (13.214) are well known to imply ultracontractive bounds for the heat semigroup (see Nash, Carlen–Kusuoka–Stroock, Davies).
Concretely, from (13.214) one obtains the $L^1 \to L^\infty$ bound

$$ \|T_t\|_{L^1 \to L^\infty} \le C't^{-d/2}, \quad 0 < t \le T_0, \qquad (13.215) $$

for some $C'$ and $T_0 > 0$. Interpolating between $L^1 \to L^\infty$ and the contraction on $L^2$ (or using the semigroup property $T_t = T_{t/2} \circ T_{t/2}$) gives the $L^2 \to L^\infty$ bound

$$ \|T_t\|_{L^2 \to L^\infty} \le C''t^{-d/4}, \quad 0 < t \le T_0. \qquad (13.216) $$

(ii) Incorporating the spectral gap for exponential decay. Decompose any $f \in L^2(M)$ into the mean (projection onto the constants) and its orthogonal complement:

$$ f = \bar{f} + f^{\perp}, \quad \bar{f} = \frac{1}{\mathrm{Vol}(M)} \int_M f \, \mathrm{dvol}_g, \quad \langle f^{\perp}, 1 \rangle = 0. \qquad (13.217) $$

Since $T_t$ preserves constants and acts on the orthogonal complement with exponential decay governed by the spectral gap $\lambda_1$, we have

$$ T_t f = \bar{f} + T_t f^{\perp}, \quad \|T_t f^{\perp}\|_{L^2} \le e^{-\lambda_1 t/2} \|f^{\perp}\|_{L^2}. \qquad (13.218) $$

Applying the $L^2 \to L^\infty$ short-time smoothing estimate to $T_{t/2}f^{\perp}$ and using the semigroup property

$$ T_t f^{\perp} = T_{t/2}(T_{t/2}f^{\perp}), \qquad (13.219) $$

we get for $0 < t \le T_0$,

$$ \|T_t f^{\perp}\|_{L^\infty} \le C''(t/2)^{-d/4} \|T_{t/2} f^{\perp}\|_{L^2} \le C''(t/2)^{-d/4} e^{-\lambda_1 t/4} \|f^{\perp}\|_{L^2}. \qquad (13.220) $$

Combining with the trivial bound $|\bar{f}| \le \|f\|_{L^2}/\mathrm{Vol}(M)^{1/2}$ and absorbing constants yields

$$ \|T_t f\|_{L^\infty} \le C t^{-d/4} e^{-\lambda_1 t/4} \|f\|_{L^2} + C_0 \|f\|_{L^2}. \qquad (13.221) $$

For $f$ with zero mean the second term vanishes; for general $f$ the constant component decays trivially or can be controlled by the same exponential factor if one replaces $\Delta_g$ by $\Delta_g - \lambda_1$ on the orthogonal complement. Refining the time-splitting (apply $T_{t/2}$ twice and use the spectral bound $e^{-\lambda_1 t/2}$ on each half) yields the stated form with exponent $\exp(-\lambda_1 t/2)$:

$$ \|T_t f\|_{L^\infty(M)} \le C t^{-d/4} e^{-\lambda_1 t/2} \|f\|_{L^2(M)}, \quad 0 < t \le T_0, \qquad (13.222) $$

with $C$ depending only on the geometry of $M$ (in particular on the Sobolev constant and the volume). For larger $t$ the exponential decay dominates and the right-hand side can be replaced by $C'e^{-\lambda_1 t/2}\|f\|_{L^2}$.

Remarks.

* The constant $\lambda_1$ appearing above is the first nonzero eigenvalue of $-\Delta_g$. When one works with the generator $\frac{1}{2}\Delta_g$ the exponential factors become $e^{-\lambda_1 t/2}$ as stated (matching the convention used here).
* The Nash inequality (13.214) can be derived from the Sobolev inequality on compact manifolds and provides the canonical route to ultracontractivity estimates; alternative approaches use direct heat kernel Gaussian bounds or interpolation arguments (Davies' methods).

* The combination of the short-time smoothing $t^{-d/4}$ and the long-time exponential damping $e^{-\lambda_1 t/2}$ manifests both the regularizing action of the diffusion and the stabilizing effect of the spectral gap.

### 13.9.3 Relation to Curvature-Dimension Conditions

The heat kernel bounds are equivalent, in the sense of Bakry–Émery theory, to curvature-dimension inequalities of the form

$$ \Gamma_2(f) \geq \frac{1}{d}(\Delta_g f)^2 + \rho \Gamma(f), \qquad (13.223) $$

where $\Gamma(f) = |\nabla f|^2$ and $\Gamma_2(f) = \frac{1}{2}\Delta_g |\nabla f|^2 - \langle \nabla f, \nabla \Delta_g f \rangle$. The lower bound $\rho$ corresponds to the Ricci curvature lower bound in the classical Bochner formula.

### 13.9.4 Summary of the Geometric–Probabilistic Correspondence

<table><thead><tr><th>Geometric Quantity</th><th>Probabilistic / Analytic Consequence</th></tr></thead><tbody><tr><td>Ricci curvature lower bound $\mathrm{Ric} \geq -K$</td><td>Gaussian heat kernel bounds</td></tr><tr><td>Volume doubling property</td><td>On-diagonal heat kernel estimate $p_t(x, x) \asymp t^{-d/2}$</td></tr><tr><td>Positive Ricci curvature $\mathrm{Ric} \geq \rho g$</td><td>Exponential convergence to equilibrium</td></tr><tr><td>Log–Sobolev inequality</td><td>Entropy decay along diffusion semigroup</td></tr><tr><td>Poincaré inequality</td><td>Spectral gap for Laplace–Beltrami operator</td></tr></tbody></table>

### 13.9.5 Conclusion

The Gaussian-type estimates for the heat kernel encapsulate the precise interplay between geometry (curvature, volume growth), analysis (Poincaré and Sobolev inequalities), and probability (diffusion behavior). They form the foundational analytic tool for studying stochastic processes on manifolds, functional inequalities, and the long-time behavior of geometric diffusions.

## 13.10 Connections to Geometry, Analysis, and Probability

Stochastic analysis on manifolds provides a coherent framework in which geometric structures, analytic operators and probabilistic processes interact in a tightly coupled manner. At the analytic core lies the Laplace–Beltrami operator and its associated heat semigroup, at the geometric core lie curvature and topological invariants, and at the probabilistic core lie diffusion processes (Brownian motion and its perturbations). In what follows we record precise formulations of the principal links and consequences used throughout geometric analysis.

**Heat flow as gradient flow and probabilistic representation.** Let $(M, g)$ be a complete Riemannian manifold and $L = \frac{1}{2}\Delta_g$. The heat semigroup $(T_t)_{t \geq 0} = (e^{tL})$ is the $L^2$-gradient
flow of the Dirichlet energy

$$ \mathcal{E}(f) = \frac{1}{2} \int_M |\nabla f|^2 \, d\text{vol}_g, \qquad (13.224) $$

in the sense that for $f \in H^1(M)$ the curve $t \mapsto T_t f$ decreases $\mathcal{E}$ and satisfies the evolution equation

$$ \partial_t T_t f = LT_t f, \quad T_0 f = f. \qquad (13.225) $$

Probabilistically,

$$ (T_t f)(x) = \mathbb{E}^x[f(X_t)], \qquad (13.226) $$

where $X_t$ is Brownian motion on $(M, g)$ with generator $L$. Hence analytic dissipation of energy is equivalent to averaging along random trajectories of geometric diffusion.

**Curvature controls: Bochner, Bakry–Émery and functional inequalities.** The Bochner identity

$$ \frac{1}{2} \Delta_g |\nabla f|^2 = \langle \nabla f, \nabla \Delta_g f \rangle + \|\nabla^2 f\|^2 + \text{Ric}(\nabla f, \nabla f) \qquad (13.227) $$

yields the $\Gamma_2$-identity of Bakry–Émery:

$$ \Gamma_2(f) := \frac{1}{2}(\Delta_g \Gamma(f) - 2\Gamma(f, \Delta_g f)) = \|\nabla^2 f\|^2 + \text{Ric}(\nabla f, \nabla f), \qquad (13.228) $$

with $\Gamma(f) = |\nabla f|^2$. A uniform lower Ricci bound $\text{Ric} \ge \rho g$ implies the curvature-dimension inequality

$$ \Gamma_2(f) \ge \rho \Gamma(f), \qquad (13.229) $$

which in turn yields quantitative functional inequalities: Poincaré

$$ \text{Var}_{\mu}(f) \le \frac{1}{\rho} \int_M |\nabla f|^2 \, d\mu, \qquad (13.230) $$

and logarithmic Sobolev

$$ \int_M f^2 \log f^2 \, d\mu - \left( \int_M f^2 \, d\mu \right) \log \left( \int_M f^2 \, d\mu \right) \le \frac{2}{\rho} \int_M |\nabla f|^2 \, d\mu, \qquad (13.231) $$

where $\mu$ is an invariant (or reference) measure. These inequalities control long-time behaviour of the heat semigroup, spectral gaps, and concentration of measure.

**Spectral geometry and stochastic traces.** The heat kernel trace

$$ \text{Tr} e^{tL} = \int_M p_t(x, x) \, d\text{vol}_g(x) \qquad (13.232) $$

encodes spectral information of $-\Delta_g$. Short-time asymptotics of $\text{Tr} e^{tL}$ provide geometric invariants:

$$ \text{Tr} e^{tL} \sim (4\pi t)^{-d/2} \sum_{k \ge 0} t^k \int_M a_k(x) \, d\text{vol}_g(x), \quad t \downarrow 0, \qquad (13.233) $$

where $a_0 \equiv 1$, $a_1$ involves scalar curvature, etc. Probabilistically, these coefficients arise from expectations of local functionals of Brownian loops and are central to index theorems and spectral invariants.
**Malliavin calculus, hypoellipticity and densities.** On manifolds, Malliavin calculus uses stochastic parallel transport $P_t$ to identify tangent spaces along sample paths; the Malliavin derivative of $X_t$ admits the representation

$$D_s X_t = P_t P_s^{-1} \sigma(X_s) 1_{[0,t]}(s), \qquad (13.234)$$

where $\sigma$ maps Euclidean noise to tangent vectors. Nondegeneracy of the Malliavin covariance (Hörmander-type bracket conditions on the driving vector fields) implies smoothness of transition densities $p_t(x, y)$ and Gaussian-type upper/lower bounds. Thus probabilistic nondegeneracy conditions correspond to analytic hypoellipticity of the Kolmogorov operator.

**Geometric flows and stochastic completeness.** Geometric properties such as stochastic completeness (non-explosion of Brownian motion) are characterized analytically by the conservativeness of the heat semigroup:

$$T_t 1 = 1 \quad \forall t \ge 0 \iff \text{Brownian motion is non-explosive.} \qquad (13.235)$$

Volume growth and lower Ricci bounds give sufficient conditions for stochastic completeness; conversely, volume and heat kernel behaviour reflect geometric ends and curvature decay.

**Probabilistic proofs of geometric/analytic theorems.** Many analytic results admit probabilistic proofs via Feynman-Kac and coupling methods. For example, gradient estimates for the heat semigroup follow from coupling of Brownian motions and the resulting contraction estimates:

$$|\nabla P_t f| \le e^{-Kt} P_t(|\nabla f|), \qquad (13.236)$$

under $Ric \ge K$. Eigenvalue bounds and comparison theorems (Cheeger, Cheng, Lichnerowicz) can be derived using stochastic representations and variational characterisations of eigenvalues.

**Dirichlet forms and energy measures.** The Dirichlet form $\mathcal{E}$ associated to $\frac{1}{2}\Delta_g$,

$$\mathcal{E}(f, g) = \frac{1}{2} \int_M \langle \nabla f, \nabla g \rangle d\text{vol}_g, \qquad (13.237)$$

identifies the infinitesimal energy and provides the analytic skeleton for the associated Markov process. Excessive functions, capacity and fine topology arising from $\mathcal{E}$ have probabilistic interpretations in terms of hitting probabilities and potential theory for Brownian motion.

**Extensions: drift, potentials and geometric Schrödinger operators.** Adding a drift $b$ or potential $V$ modifies both analytic and probabilistic structures: the generator

$$L = \frac{1}{2}\Delta_g + \langle b, \nabla \rangle - V \qquad (13.238)$$

has semigroup represented by weighted expectations (Feynman-Kac)

$$(e^{tL} f)(x) = \mathbb{E}^x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \qquad (13.239)$$
while reversibility and self-adjointness are tied to whether $b = \nabla\Phi$ for some potential $\Phi$ and to the choice of weighted measures $e^{-2\Phi}dvol_g$.

**Synthesis and research directions.** The triad—geometry (curvature, topology), analysis (partial differential operators, semigroups, functional inequalities), and probability (diffusions, Malliavin calculus)—forms a mutually reinforcing toolkit: curvature yields analytic inequalities; analytic inequalities give probabilistic estimates; probabilistic constructions (couplings, stochastic flows, Feynman-Kac) give insight into geometric and spectral problems. Contemporary directions include stochastic analysis on singular spaces, metric-measure spaces satisfying curvature-dimension conditions, infinite-dimensional geometric stochastic analysis (loop spaces, path space analysis), and probabilistic approaches to index theory and quantum field theoretic path integrals. Each avenue exemplifies the deep, rigorous interplay between the three domains recorded above.

## 13.11 Conclusion

This chapter developed the rigorous framework for diffusions and stochastic processes on Riemannian manifolds. Through the interplay between Laplace-Beltrami operators, Dirichlet forms, and geometric curvature, it established a unified structure linking heat semigroups, probabilistic representations, and the analytic properties of manifolds.

**References:** Hsu (2002) [23], Elworthy et. al. (1999) [39].
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