数理解析研究所講究録
第 1837 巻 2013 年 102-113

102

Long time asymptotic problems for stochastic
optimal control and related variational problems
Naoyuki Ichihara*
Graduate School of Engineering,
Hiroshima University

Abstract

In this note we present some recent results on the large time behavior of solutions to viscous Hamilton-Jacobi equations arising in stochastic control. Our
equations possess superlinear nonhnearity in gradients, and solutions are unbounded on the whole Euclidean space. We prove that, as the time tends toinfinity, the solution approaches to a steady state in a suitable sense. We alsoestablish a variational representation formula for the limit.

1

Introduction

Let us consider semihnear parabolic equations of the form$\partial_{t}u-\frac{1}{2}tr(a(x)D^{2}u)+H(x, Du)=0$

in$(0, \infty)\cross \mathbb{R}^{N}$

(1.1)

,

. We are concerned with
, and
where
the large time behavior of solutions of (1.1). It turns out under suitable assumptions
on $a=(a_{ij}(x)),$ $H=H(x,p)$ , and initial datum $u(0, \cdot)$ , that the solution $u=u(t, x)$of (1.1) approaches as $tarrow\infty$ to a function of the form $\lambda t+\phi(x)+c$ for some realwith $\phi(0)=0$ . More precisely, one can, and function $\phi=\phi(x)$ on
constants
prove the following convergence:$\partial_{t}u=\partial u/\partial t,$$Du=(\partial u/\partial x_{i})$$D^{2}u=(\partial^{2}u/\partial x_{i}\partial x_{j})$$\mathbb{R}^{N}$$\lambda,$$c$$u(t, x)-(\lambda t+\phi(x)+c)arrow 0$

in$C(\mathbb{R}^{N})$as$Tarrow\infty$

.

(1.2)

E–mail: $naoyukiQhiroshima-u$ . ac. jp. Supported in part by JSPS KAKENHI Grant Number

24740089.

103
Here, convergence “instands for locally uniform convergence in. We callthe triplet
asymptotic solution if $\lambda t+\phi(x)+c$ solves (1.1). Any asymptotic
solution should satisfy the stationary equation$C(\mathbb{R}^{N})^{\mathfrak{n}}$$\mathbb{R}^{N}$$(\lambda, \phi, c)$$\lambda-\frac{1}{2}tr(a(x)D^{2}\phi)+H(x, D\phi)=0$

Finding a pairin$\mathbb{R}^{N},$$\phi(0)=0$

.

(1.3)

satisfying (1.3) is called ergodic problem. Remark that and in(1.2) are specified from the stationary equation (1.3), whereas the constant needs tobe determined from the evolutionary equation (1.1). Asymptotic problems of this typehave been largely studied in various settings. We refer to [1, 2, 5, 12, 13] for recent
results from the PDE viewpoint, and to [3, 4, 6, 8, 9, 10, 11] from the probabilistic
viewpoint, especially, in connection with mathematical finance.
In this note, we concentrate on a more specific equation: we consider the Cauchy$(\lambda,\phi)$$\lambda$$\phi$$c$problem( $CP$ )$\{\begin{array}{ll}\partial_{t}u-\frac{1}{2}\triangle u+\frac{1}{m}|Du|^{m}=f in (0, +\infty)\cross \mathbb{R}^{N},u|_{t=0}=g on \{0\}\cross \mathbb{R}^{N},\end{array}$where $m,$ , and$f$

are assumed to satisfy the following conditions:

$g$

(Hl)

$m>1.$(H2)$f\in C^{2}(\mathbb{R}^{N})$, and there exist some $C>0$ and $\beta>0$ such that

$C^{-1}|x|^{\beta}-C\leq f(x)\leq C(|x|^{\beta}+1)$(H3)$g\in C(\mathbb{R}^{N})$is bounded below on,$|Df(x)|\leq C(|x|^{\beta-1}+1)$,$x\in \mathbb{R}^{N}.$$\mathbb{R}^{N}.$In the first half of this note, we discuss, according to [9], the large time behavior ofsolutions to ( $CP$ ). It holds convergence (1.2) for someunder (Hl) $-(H3)$ . In thesecond half, we study a variational representation formula for the limit , which seems
to be new to the best of our knowledge.
Equation ( $CP$ ) naturally appears in the stochastic control theory. Let us consider
the following minimizing problem$(\lambda, \phi, c)$$c$

Minimize$J(T, x;\xi)$ $:=E[ \int_{0}^{T}(\frac{1}{m}*|\xi_{t}|^{m^{*}}+f(X_{t}^{\xi}))dt+g(X_{T}^{\xi})],$

subject to$X_{t}^{\xi}=x- \int_{0}^{t}\xi_{s}ds+W_{t},$$t\geq 0,$

where $m^{*};=m/(m-1)>1$ , and $W=(W_{t})$ denotes an $N$ -dimensional Brownian
motion defined on a filtered probability space. The control process$(\Omega,\mathcal{F}, P;(\mathcal{F}_{t}))$104
is taken from the admissible class-progressively measurable processes$\xi=(\xi_{t})$$(\mathcal{F}_{t})$which is defined as the collection of allinsuch that$\mathcal{A}_{T}$$\mathbb{R}^{N}$$\xi=(\xi_{t})$$E^{x}[ \int_{0}^{T}(|\xi_{t}|^{m^{*}}+|X_{t}^{\xi}|^{\beta})dt]<\infty, x\in \mathbb{R}^{N}$

where$\beta$(1.4),

is the constant in (H2). Then, we see that the value function(1.5)$u(T, x) := \inf_{\xi\in \mathcal{A}_{T}}J(T,x;\xi)$is a classical solution of ( $CP$ ).
This note is organized as follows. In the next section, we survey some results
obtained in [9]. In Section 3, we discuss a variational representation formula for theconstant in (1.2).
$c$2

Convergence of solutions

We begin with the solvability of ( $CP$ ).
Theorem 2.1. Let (Hl) $-(H3)$ hold. Then,of ( $CP$ ) in the class$u$defined by (1.5) is the minimal solution$\Phi:=\{u\in C^{1,2}((0, \infty)\cross \mathbb{R}^{N})\cross C([0,\infty)\cross \mathbb{R}^{N})| \inf u>-\infty, T>0\}.$$[0,T]xR^{N}$

Proof. The proof is based on the verification theorem. See [9, Theorem 2.1] for details.
$\square$As the limiting equation of ( $CP$ ), we derive the ergodic problem$\lambda-\frac{1}{2}\Delta\phi+\frac{1}{m}|D\phi|^{m}=f$

Recall that the unknown isin the following sense.

in$(\lambda, \phi)\in \mathbb{R}\cross C^{2}(\mathbb{R}^{N})$$\mathbb{R}^{N},$$\phi(0)=0$.

( $EP$ ). Equation ( $EP$ ) has a unique solution

Theorem 2.2. Let (Hl) $-(H3)$ hold. Then, there exists a unique solutionof ( $EP$ ) such that
. Moreover, there exists some $C>0$ suchthat the solution satisfies the following estimate:$(\lambda^{*}, \varphi)\in$$\inf_{R^{N}}\varphi>-\infty$$\mathbb{R}\cross C^{2}(\mathbb{R}^{N})$$\varphi$$C^{-1}|x|^{(\beta/m)+1}-C\leq\varphi(x)\leq C(|x|^{(\beta/m)+1}+1) , x\in \mathbb{R}^{N}.$Proof. See [9, Theorem 2.2].

$\square$105
is necessary to derive the uniqueness ofRemark 2.3. The condition
solution. Indeed, there exist infinitely may pairs
satisfying ( $EP$ ) if we do not put
this condition.
$\inf_{\mathbb{R}^{N}}\varphi>-\infty$$(\lambda, \phi)$Letthe solution$(\lambda^{*}, \varphi)$$u$be the unique solution of ( $EP$ ) given in Theorem 2.2. Then, we see thatfor someof ( $CP$ ) converges to an asymptotic solution$c\in \mathbb{R}.$$(\lambda^{*}, \varphi, c)$be the solutions of ( $CP$ ) andTheorem 2.4. Let (Hl) $-(H3)$ hold. Let and. Then, there exists a constant such that
( $EP$ ), respectively. Assume that$u$$(\lambda^{*}, \varphi)$$\beta\geq m^{*}$$c$$u(T, \cdot)-(\lambda^{*}T+\varphi(\cdot)+c)arrow 0$

in$C(\mathbb{R}^{N})$as$Tarrow\infty$

.

(2.1)

Remark 2.5. Under (Hl) $-(H3)$ , we can prove thatin$\frac{u(T,\cdot)}{T}arrow-\lambda^{*}$as$C(\mathbb{R}^{N})$$Tarrow\infty.$

However, we do not know, in general, if (2.1) is valid without assuming$\beta\geq m^{*}.$In the rest of this section, we give a sketch of the proof for Theorem 2.4. We
refer to [9, Section 5.2] for a complete proof. Let be the solution of ( $CP$ ) defined. We setbe the solution of ( $EP$ ) such thatby (1.5), and let$w(T, x)$ $:=u(T, x)-(\varphi(x)+\lambda^{*}T)$ for. The goal is to prove that
$Tarrow\infty$ .
$w(T, \cdot)$ converges in
to a constant as
Observe that is a solution of$u$$\inf_{\mathbb{R}^{N}}\varphi>-\infty$$(\lambda^{*}, \varphi)$$(T, x)\in(0, \infty)\cross \mathbb{R}^{N}$$C(\mathbb{R}^{N})$$w$$\partial_{t}w-A^{\varphi}w+H_{\varphi}(x, Dw)=0$

with $w(0, \cdot)=g-\varphi$ inby$\mathbb{R}^{N}$, where$A^{\varphi}$

in

(2.2)

$(0, \infty)\cross \mathbb{R}^{N}$is the second order differential operator given$A^{\varphi} := \frac{1}{2}\triangle-|D\varphi(x)|^{m-2}D\varphi(x)\cdot D,$

and$H_{\varphi}(x,p)$

is defined by.

$H_{\varphi}(x,p) := \frac{1}{m}|p+D\varphi(x)|^{m}-\frac{1}{m}|D\varphi(x)|^{m}-|D\varphi(x)|^{m-2}D\varphi(x)\cdot p$

Notice thatLet
ential equation$H_{\varphi}\geq 0$

in$X^{\varphi}=(X_{t}^{\varphi})_{t\geq 0}$(2.3)

since the mapping $p\mapsto(1/m)|p|^{m}$ is convex.
-diffusion, that is, the solution of the stochastic differbe the$\mathbb{R}^{2N}$$A^{\varphi}$$dX_{t}^{\varphi}=-|D\varphi(X_{t}^{\varphi})|^{m-2}D\varphi(X_{t}^{\varphi})dt+dW_{t}, t\geq 0.$

Note that$X^{\varphi}$

is ergodic with an invariant probability measure $\mu=\mu(dx)$ such thatfor all $l>0$ (see [9, Proposition 4.13]).

$\int_{R^{N}}.|x|^{l}\mu(dx)<\infty$

106
Lemma 2.6. Let
be the$X^{\varphi}=(X_{t}^{\varphi})$be the unique solution of ( $EP$ ) given in Theorem 2.2, and let-diffusion. Then,$(\lambda^{*}, \varphi)$$A^{\varphi}$$w(T+S, x)\leq E^{x}[w(T,X_{S}^{\varphi})], T, S\geq 0, x\in \mathbb{R}^{N}.$Proof. In view of Ito’s formula to$w(T+S-t, X_{t}^{\varphi})$and equation (2.2), we see that$w(T+S-S\wedge\tau_{R}, X_{S\wedge\tau}^{\varphi}R)-w(T+S, X_{0}^{\varphi})$$= \int_{0}^{S\wedge \mathcal{T}R}(-\partial_{t}w+A^{\varphi}w)(T+S-t, X_{t}^{\varphi})dt+\int_{0}^{S\tau}\wedge RDw(T+S-t, X_{t}^{\varphi})dW_{t}$$\geq\int_{0}^{s\wedge R}\tau Dw(T+S-t, X_{t}^{\varphi})dW_{t},$where $\tau_{R}:=\inf\{t>0||X_{t}^{\varphi}|\geq R\}$ . Taking expectation, we have$w(T+S, x)\leq E^{x}[w(T+S-S\wedge\tau_{R,\wedge R}X_{S\tau}^{\varphi})].$

for some $C,$ $q>1$ , andSince $|w(t, x)|\leq C(1+|x|^{q})$ in
is uniformly integrable, we obtain the desired estimate after sending $Rarrow\infty.$$\{|X_{S\wedge\tau}^{\varphi}R|^{q};R>1\}$$[0, T+S]\cross \mathbb{R}^{N}$$\square$Proposition 2.7. The family $\{w(T, \cdot)|T>1\}$ is uniformly bounded from above on, then it is also uniformly$:=\{x\in \mathbb{R}^{N}||x|\leq R\}$ for any $R>0$ . Moreover, if
bounded from below on$\beta\geq m^{*}$$\overline{B}_{R}$$\overline{B}_{R}.$Proof. Let$X^{\varphi}=(X_{t}^{\varphi})_{t\geq 0}$

be the$A^{\varphi}$diffusion. Then, in view of Lemma 2.6, we seethat$w(T, x) \leq E^{x}[(g-\varphi)(X_{T}^{\varphi})]arrow\int_{R^{N}}(g-\varphi)(y)\mu(dy)<\infty$

ae$Tarrow\infty.$, we see that $w(T, \cdot)$ is bounded above
Since the convergence above is uniform inuniformly in $T>1.$on. Set $v(T, x)$ $:=(1-e^{-\delta T})\varphi(x)+\lambda T+q(T)$To get a lower bound, we aesume
for some $\delta>0$ and $q\in C^{1}([0, \infty))$ that will be determined later. Then, noting$\varphi(x)\leq K(1+|x|^{(\beta/m)+1})$ infor some $K>0$ and observing $\beta\geq(\beta/m)+1$ in view
, we haveof$\overline{B}_{R}$$\overline{B}_{R}$$\beta\geq m^{*}$$\mathbb{R}^{N}$$\beta\geq m^{*}$$\partial_{t}v+F[v|\leq e^{-\delta T}\delta\varphi+\lambda+q’+(1-e^{-\delta T})F[\varphi]+e^{-\delta T}F[0]$$\leq e^{-\delta T}(\delta K-c_{1})|x|^{\beta}+q’+e^{-\delta T}(2\delta K+|\lambda|+C_{1})$for some$c_{1},$$C_{1}>0$$C_{1})(1-e^{-\delta T})$

. We now choose. Then,$\partial_{t}v+F[v]\leq 0$$\delta$in$:=c_{1}/K$

and $q(T)$ $:= \inf_{R^{N}}g-\delta^{-1}(2\delta K+|\lambda|+$. In particular,
and $v(0, \cdot)\leq g$ in$(0, \infty)\cross \mathbb{R}^{N}$$\mathbb{R}^{N}$

107
is a subsolution of ( $CP$ ). Applying the comparison principle ([9, Proposition 3.6]),
we obtain $v\leq u$ in. This infers that $-e^{-\delta T}\varphi(x)+q(T)\leq w(T, x)$ for all. Since $\inf_{T>1}q(T)>-\infty$ , we conclude that $w(T, \cdot)$ is bounded
below onuniformly in $T>1$ . Hence, we have completed the proof.
$v$$(0, \infty)\cross \mathbb{R}^{N}$$(T, x)\in(O, \infty)\cross \mathbb{R}^{N}$$\overline{B}_{R}$$\square$

Let$\Gamma$be the totality of all -limits of $\{w(T, \cdot)|T>1\}$ in$\omega$$\Gamma$$:= \{w_{\infty}\in C(\mathbb{R}^{N})|\lim_{jarrow\infty}w(T_{j}, \cdot)=w_{\infty}$in$C(\mathbb{R}^{N})$$C(\mathbb{R}^{N})$for some$\lim_{jarrow\infty}T_{j}=\infty\}.$In view of Proposition 2.7 and the standard gradient estimate forfamily $\{w(T, \cdot)|T>1\}$ is pre-compact in. In particular,
We are now in position to complete the proof of Theorem 2.4.
$C(\mathbb{R}^{N})$Proof of Theorem 2.4. It suffices to prove that, namely,$w$, we see that the$\Gamma\neq\emptyset.$for some$\Gamma=\{c\}$$c\in \mathbb{R}$that any element of is constant. Let, i.e.,
for some diverging sequence. By Lemma 2.6, we see that$\Gamma$$w_{\infty}\in\Gamma$$jarrow\infty$. We first show

$w(T_{j}, \cdot)arrow w_{\infty}$in$C(\mathbb{R}^{N})$as$\{T_{j}\}$$w(T+S, x)\leq E^{x}[w(T, X_{S}^{\varphi})], T, S\geq 0, x\in \mathbb{R}^{N}$

.

(2.4)

Take $S:=T_{j}-T$ and send $jarrow\infty$ . Then, we have$w_{\infty}(x) \leq\int w(T, y)\mu(dy)$.

Choosing $T:=T_{j}$ and letting $jarrow\infty,$$w_{\infty}(x) \leq\int w_{\infty}(y)\mu(dy)$Taking the $\sup$ over$x\in \mathbb{R}^{N}$.

, we obtain$0 \leq\int(w_{\infty}(y)-\sup_{\mathbb{R}^{N}}w_{\infty})\mu(dy)\leq 0.$

the last estimate and the fact that $supp\mu=\mathbb{R}^{N}$ , we conclude thatin
. Hence,is constant in
We next show that consists of a single element. Suppose that there exist two
diverging sequencesandsuch that $w(T_{j}, \cdot)arrow c_{1}$ and $w(S_{j}, \cdot)arrow c_{2}$ in
as $jarrow\infty$ for some
. We choose $:=S_{j}-T$ and $T:=T_{k}$ in (2.4),$karrow\infty$
and let $jarrow\infty$ andin this order. Then,$\mathbb{R}om$$w_{\infty}=$$\mathbb{R}^{N}$$\mathbb{R}^{N}.$$\sup_{\mathbb{R}^{N}}w_{\infty}$$w_{\infty}$$\Gamma$$\{T_{j}\}$$C(\mathbb{R}^{N})$$\{S_{j}\}$$S$$c_{1},$ $c_{2}\in \mathbb{R}$$c_{2} \leq\lim_{karrow\infty}\int w(T_{k}, y)\mu(dy)=\int c_{1}\mu(dy)=c_{1}.$. Changing the role ofThus,$c_{2}\leq c_{1}$$c_{1}=c_{2}$, and therefore $\Gamma=\{c\}$ for some$\{T_{j}\}$and$c\in \mathbb{R}.$$\{S_{j}\}$, we also have$c_{1}\leq c_{2}$. Hence,$\square$

108

3

$A$

representation formula

In this section, we discuss the dependence of in (2.1) with respect to the initial
be the solutions of ( $CP$ ) and ( $EP$ ), respectively. As infunction . Let andthe previous section, we set
$c$

$u$

$g$

$(\lambda^{*}, \varphi)$$w(T, x):=u(T, x)-(\lambda^{*}T+\varphi(x)) , T\geq 0, x\in \mathbb{R}^{N}$

.

(3.1)

satisfies (2.2) with $w(0, \cdot)=g-\varphi$ . In the rest of this section, we set$\eta:=w(0, \cdot)$ , which is viewed as a small perturbation of stationary state. In view ofTheorem 2.4, we can prove the following theorem.
Then,

$w$

$\varphi$, there exists a real constant $c=c(\eta)$ such that
Theorem 3.1. For any$w(t, \cdot)arrow c$ in. Moreover, let $\mu=\mu(dx)$ be the invariant probabilityasmeasure for the -diffusion. Then, the function$\eta\in C_{b}(\mathbb{R}^{N})$$tarrow\infty$$C(\mathbb{R}^{N})$$A^{\varphi}$$t \mapsto\langle w(t, \cdot),\mu\rangle:=\int_{R^{N}}w(t, x)\mu(dx)$is non-increasing. In particular,$c( \eta)=\inf_{t>0}\langle w(t, \cdot), \mu\rangle=\lim_{tarrow\infty}\langle w(t, \cdot),\mu\rangle.$

In what follows, we assume
taken from the Banach space$\eta\in C_{b}(\mathbb{R}^{N})$and regard $c=c(\eta)$ as a functional of, where$(C_{b}(\mathbb{R}^{N}), \Vert \Vert_{\infty})$$\Vert\eta\Vert_{\infty}$$:= \sup_{R^{N}}|\eta|.$

Proposition 3.2. Let $c=c(\eta)$ be the constant given in Theorem 3.1. Then,satisfies the following properties:(a) $c(O)=0$ and $c(\eta+a)=c(\eta)+a$ for any
and
(b)
in
.
implies(c)
for all.
(d)
is concave, i.e.,
for all$\mathbb{R}^{N}$$c(\eta_{1})\leq c(\eta_{2})$$|c(\eta_{1})-c(\eta_{2})|\leq\Vert\eta_{1}-\eta_{2}\Vert_{\infty}$$\eta_{1},$$\eta_{2}\in C_{b}(\mathbb{R}^{N})$$c(\delta\eta_{1}+(1-\delta)\eta_{2})\geq\delta c(\eta_{1})+(1-\delta)c(\eta_{2})$$c$$c(\eta)$$a\in \mathbb{R}.$$\eta\in C_{b}(\mathbb{R}^{N})$$\eta_{1}\leq\eta_{2}$$\eta$$\eta_{1},$$\eta_{2}\in C_{b}(\mathbb{R}^{N})$and $\delta\in[0,1].$

Proof. (a). Let

be the nonlinear semigroup associated with (2.2), that is, for, we seteach, where denotes the unique solutionof (2.2) with $w(0, \cdot)=\eta$ . Then, by the uniqueness of solution, it is easy to see thatand $T_{t}(\eta+a)=T_{t}\eta+a$ . In particular, $c(O)=0$ and $c(\eta+a)=c(\eta)+a.$(b). Sincein view of comparison, we obtainafter sending$(T_{t})_{t\geq 0}$$\eta\in C_{b}(\mathbb{R}^{N})$$T_{t}\eta$$:=w(t, \cdot)\in C_{b}(\mathbb{R}^{N})$$w$$T_{t}0\equiv 0$$T_{t}(\eta_{1})\leq T_{t}(\eta_{2})$$tarrow\infty.$$c(\eta_{1})\leq c(\eta_{2})$109(c). Set(b), we see that$:=\eta_{2}+\Vert\eta_{1}-\eta_{2}\Vert_{\infty}$$\eta$. Note that$\eta_{1}\leq\eta$in$\mathbb{R}^{N}$. Taking into acount (a) and$T_{t}\eta_{1}\leq T_{t}\eta=T_{t}(\eta_{2}+\Vert\eta_{1}-\eta_{2}\Vert_{\infty})=T_{t}\eta_{2}+\Vert\eta_{1}-\eta_{2}\Vert_{\infty}.$, we obtain
Letting
we obtain the desired inequality.
(d). In view of the convexity ofa subsolution of (2.2) with $w(0, \cdot)$
we haveconcavity of$tarrow\infty$. Changing the role of$c(\eta_{1})\leq c(\eta_{2})+\Vert\eta_{1}-\eta_{2}\Vert_{\infty}$$H_{\varphi}(x,p)$$\eta_{1}$and$\eta_{2},$in , we see thatis
. By the comparison theorem,. Letting $tarrow\infty$ , we obtain the$\delta T_{t}(\eta_{1})+(1-\delta)T_{t}(\eta_{2})$$p$$:=\delta\eta_{1}+(1-\delta)\eta_{2}$$\delta T_{t}(\eta_{1})+(1-\delta)T_{t}(\eta_{2})\leq T_{t}(\delta\eta_{1}+(1-\delta)\eta_{2})$$c.$$\square$

We now derive a variational formula for. Let
be a given filtered$N$
probability space on which is defined an -dimensional standard-Brownian motion$W=(W_{t})_{t\geq 0}$ . Let
denote the totality of-progressively measurable processes
with values in. For each $T>0,$, and a given initial law,
we define the stochastic process
as the solution to the stochastic differential
equation$c(\eta)$$(\Omega, \mathcal{F}, P;(\mathcal{F}_{t}))$$(\mathcal{F}_{t})$$\mathcal{A}_{T}$$(\mathcal{F}_{t})$$\mathbb{R}^{N}$$\xi=(\xi_{t})_{0\leq t\leq T}$$\xi\in \mathcal{A}_{T}$$X^{\xi}=X^{\xi}$$dX_{t}^{\xi}=-\xi_{t}dt-|D\varphi(X_{t}^{\xi})|^{m-2}D\varphi(X_{t}^{\xi})dt+dW_{t}, 0\leq t\leq T$Let$H_{\varphi}=H_{\varphi}(x,p)$

.

(3.2)

be the function defined by (2.3), and set$L(x,p):= \sup_{p\in \mathbb{R}^{N}}(\xi\cdot p-H_{\varphi}(x,p)) , (x,p)\in \mathbb{R}^{2N}.$

Note that
(Ll)
(L2)
(L3)

$L$

satisfies the following properties:$L\in C^{2}(\mathbb{R}^{N}\cross(\mathbb{R}^{N}-\{0\}))$.

for all$L(x, \xi)$ is strictly convex and superlinear with respect to$\min\{L(x, \xi)|\xi\in \mathbb{R}^{N}\}=0$$x\in \mathbb{R}^{N}.$$\xi$

for all$x\in \mathbb{R}^{N}.$For given, whereis the set of Borel probability meaeures, we consider the minimization problem
on$\mu,$$\nu\in \mathcal{M}_{1}$$\mathcal{M}_{1}=\mathcal{M}_{1}(\mathbb{R}^{N})$$\mathbb{R}^{N}$

Minimize$J_{T}(\mu, v;\xi)$subject toRecall that
, we set$X^{\xi}=(X_{t}^{\xi})$$:=E[ \int_{0}^{T}L(X_{t}^{\xi}, \xi_{t})dt]$$P(X_{0}^{\xi})^{-1}=\mu,$$P(X_{T}^{\xi})^{-1}=v,$$\xi\in \mathcal{A}_{T}.$is governed by (3.2). Furthermore, for each $T>0$ and$\mathcal{M}_{1}$$\mathcal{A}_{T}(\mu, \nu):=\{\xi\in \mathcal{A}_{T}|P(X_{0}^{\xi})^{-1}=\mu, P(X_{T}^{\xi})^{-1}=\nu\},$$V_{T}( \mu, v):=\inf\{J(\mu, \nu;\xi)|\xi\in \mathcal{A}_{T}(\mu, \nu)\},$$V( \mu, \nu):=\inf\{V_{T}(\mu, \nu)|T>0\}.$$\mu,$$\nu\in$110
We set $V_{T}(\mu, \nu):=+\infty$ if(3.1) can be written as$\mathcal{A}_{T}(\mu, \nu)=\emptyset$. Under this notation, function$w$

defined by$w(T, x)= \inf\{V_{T}(\delta_{x}, \nu)+\langle\eta, \nu\rangle|\nu\in \mathcal{M}_{1}\},$

where$\delta_{x}$stands for the unit distribution concentrated on$x\in \mathbb{R}^{N}.$Theorem 3.3. Let $c=c(\eta)$ be the constant given in Theorem 3.1. Then, for any, one has$\eta\in C_{b}(\mathbb{R}^{N})$$c( \eta)=\inf\{V(\mu, \nu)+\langle\eta, \nu\rangle|\nu\in \mathcal{M}_{1}\},$

where

$\mu$

denotes the invariant probability measure for the

Proof. Fix any$\eta\in C_{b}(\mathbb{R}^{N})$and$\nu\in \mathcal{M}_{1}$. Then, for any

such that $V_{T}(\mu, \nu)<V(\mu, \nu)+\epsilon$ . By the definition ofsuch that$V_{T}$$A^{\varphi}$-diffusion.

$\epsilon>0$, there exists a $T>0$, we can find a$\xi\in \mathcal{A}_{T}(\mu, \nu)$$E[ \int_{0}^{T}L(X_{t}^{\zeta},\xi_{t})dt]<V_{T}(\mu, \nu)+\epsilon.$

In view of Theorem 3.1, we have$c(\eta)\leq\langle w(T, \cdot),$ $\mu\rangle\leq E[\int_{0}^{T}L(X_{t}^{\xi},\xi_{t})dt+\eta(X_{T}^{\xi})]<V(\mu, \nu)+\langle\eta,$

Letting$\langle\eta,$$\epsilonarrow 0$

and then taking the inf over$\nu\in \mathcal{M}_{1}$$\nu\rangle+2\epsilon.$, we obtain $c( \eta)\leq\inf\{V(\mu, \nu)+$$\nu\rangle|\nu\in \mathcal{M}_{1}\}.$

We next prove the opposite inequality. Fix an arbitrary $T>0$ . We consider the$:=D_{p}H(x, Dw(T-t, x))$ and define the diffusion process
feedback control$X=X^{T}$ by$\xi_{T}(t, x)$$dX_{t}=-\xi_{T}(t, X_{t})dt-|D\varphi(X_{t})|^{m-2}D\varphi(X_{t})dt+dW_{t}, 0\leq t\leq T,$

with $P(X_{0})^{-i}=\mu$ . Then, by Ito’s formula and the definition of$H_{\varphi}$, we easily see that$\langle w(T, \cdot),\mu\rangle=E[\int_{0}^{T}L(X_{t},\xi_{T}(t, X_{t}))dt+\eta(X_{T})].$

Setting $\nu^{T}:=P(X_{T})^{-1}$ , we obtain$\langle w(T, \cdot),\mu\rangle\geq V_{T}(\mu, \nu^{T})+\langle\eta, v^{T}\rangle\geq\inf\{V(\mu, \nu)+\langle\eta, \nu\rangle|\nu\in \mathcal{M}_{1}\}.$

Since $T>0$ is arbitrary, we have the opposite inequality. Hence, we have completed
the proof.

$\square$111
Now, we consider the case where $m=2$ . In this case, we have $H_{\varphi}(x,p)=(1/2)|p|^{2}.$In particular, satisfies the equation$w$

in$\partial_{t}w-\frac{1}{2}\triangle w+D\varphi\cdot Dw+\frac{1}{2}|Dw|^{2}=0$

We set $v:=e^{-w}$ . Then,$v$

is a solution of the linear equation$\partial_{t}v-\frac{1}{2}\triangle v+D\varphi\cdot Dv=0$

with $v(0, \cdot)=e^{-\eta}$ ingovemed by$\mathbb{R}^{N}$$(0, \infty)\cross \mathbb{R}^{N}.$. Note that

$v$

in

$(0, \infty)\cross \mathbb{R}^{N}$is written as $v(T, x)=E_{x}[e^{-\eta(X_{T})}]$ , where $X$ is$dX_{t}=-D\varphi(X_{t})dt+dW_{t}.$Since $X$ is ergodic with invariant probability measure $\mu(dx)$ $:=e^{-2\varphi(x)}dx$ , we have$v(T, x)=E_{x}[e^{-\eta(X_{T})}] arrow\int_{\mathbb{R}^{N}}e^{-\eta(x)}\mu(dx)$Thus,$c(\eta)$as $Tarrow\infty.$

in Theorem 3.1 can be written as$c( \eta)=-\log\int_{\mathbb{R}^{N}}e^{-\eta(y)}\mu(dy) , \eta\in C_{b}(\mathbb{R}^{N})$

.

(3.3)

Taking into account this observation, we have the following representation formulafor$c(\eta)$.

Theorem 3.4. Assume that $m=2$ . Let $H(\nu|\mu)$ be the “relative entropy“ defined by$H(v| \mu):=\int_{\mathbb{R}^{N}}\log\frac{d\nu}{d\mu}(x)\nu(dx) , \nu\ll\mu,$where $H(\nu|\mu)$ $:=+\infty$ if$\nu$

is singular to$\mu$. Then,$c( \eta)=\min\{\langle\eta, \nu\rangle+H(\nu|\mu)|\nu\in \mathcal{M}_{1}\}, \mu:=e^{-2\varphi}dx.$

Moreover, the minimum is attained when

Proof. Let$\nu\in \mathcal{M}_{1}$$v=e^{-\eta}d\mu/\langle e^{-\eta},$ $\mu\rangle.$be such that $\nu\ll\mu$ , and set $p:=d\nu/d\mu$ . Then, for any$\eta\in C_{b}(\mathbb{R}^{N})$,we have$c(\eta)-\langle\eta, \nu\rangle\leq H(\nu|\mu)$.

Indeed, in view of (3.3), we see that the above inequality is equivalent to say that.

$\exp(\int_{\mathbb{R}^{N}}\{-\eta(x)-\log p(x)\}\nu(dx))\leq\int_{\mathbb{R}^{N}}e^{-\eta(x)}\mu(dx)$But this inequality is true in view of Jensen’s inequality. Hence, we obtain$c(\eta)\leq\langle\eta, v\rangle+H(\nu|\mu) , v\in \mathcal{M}_{1}.$Note that the equality holds if and only $if-\eta(x)-\log p(x)$ is constant. This implies
that should be of the form
. Hence, we have completed the$p$proof.

$p(x)=e^{-\eta(x)}/\langle e^{-\eta},$$\mu\rangle$$\square$112

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