Regularity of Schroedinger's functional equation and mean field PDEs for h-path processes

TL;DR

This paper investigates Schroedinger's functional equation and its solutions, demonstrating their measurability, connection to mean field PDEs, and interpretation as an Euler equation in stochastic optimal transportation.

Contribution

It establishes the measurability of solutions and drifts, links Schroedinger's equation to mean field PDEs, and interprets it as an Euler equation in stochastic optimal transport.

Findings

Solution of Schroedinger's functional equation is measurable in space, kernel, and marginals.

Drift vector of the h-path process depends measurably on space, time, and marginals.

Schroedinger's functional equation is the Euler equation of a stochastic optimal transportation problem.

Abstract

E. Schroedinger proposed the equation to find the statistical property of a quantum particle on a finite time interval. It is called "Schroedinger's functional equation". Given probability distributions of a particle at initial and terminal times, it determines the joint distribution of a quantum particle at initial and terminal times so that a particle is Markovian. S. Bernstein generalized Schroedinger's idea and introduced the so-called Bernstein processes which are also called reciprocal processes or one-dimensional Markov random fields. The theory of stochastic differential equation for Schroedinger's functional equation was given by B. Jamison. The solution is Doob's h-path process with given two end point marginals. We show that the solution of Schroedinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the marginals satisfy a class of mean field PDE systems of which the coefficients are measurable function of space, time and marginal. We also show that Schroedinger's functional equation is the Euler equation of a stochastic optimal transportation problem.

Equations

$$\{\begin{array}{ll}{\mu }_1(d{x}_1)={\nu }_1(d{x}_1){\int }_{S}q(x_1,x_2){\nu }_2(d{x}_2),& \\ & \\ {\mu }_2(d{x}_2)={\nu }_2(d{x}_2){\int }_{S}q(x_1,x_2){\nu }_1(d{x}_1)& \end{array}$$

$$u_{i}(x_{i}):=\log\biggl{(}\int_{S}q(x_{1},x_{2})\nu_{j}(dx_{j})\biggr{)},\quad i,j=1,2,i\neq j.$$

$${\mu }_i(dx)=\exp (u_i(x)){\nu }_i(dx),i=1,2.$$

$$\exp (u_i(x_i))={\int }_{S}q(x_1,x_2)\exp (-u_j(x_j)){\mu }_j(d{x}_j),{\mu }_i(d{x}_i)-a.s..$$

$${\nu }_i(dx)={\nu }_i(dx;q,{\mu }_1,{\mu }_2),u_i(x)=u_i(x;q,{\mu }_1,{\mu }_2),i=1,2.$$

$${\nu }_1{\nu }_2=C{\nu }_1\cdot C^{-1}{\nu }_2.$$

$${\nu }_1(A_{n_0({\mu }_1,{\mu }_2)})={\nu }_2(A_{n_0({\mu }_1,{\mu }_2)}),$$

$${\nu }_i(dx;\cdot ,\cdot ,\cdot ):C(S\times S)\times \mathcal{P}(S)\times \mathcal{P}(S)\mapsto \mathcal{M}(S),$$

$$u_i:C(S\times S)\times \mathcal{P}(S)\times \mathcal{P}(S)\mapsto C(S),$$

$${\nu }_i(dx;\cdot ,\cdot ,\cdot ):C(S\times S)\times \mathcal{P}(S)\times \mathcal{P}(S)\mapsto \mathcal{M}(S),$$

$$u_i:S\times C(S\times S)\times \mathcal{P}(S)\times \mathcal{P}(S)\mapsto \mathbf{R}\cup \{\infty \}.$$

$$dX(t)$$

$$PX(0{)}^{-1}$$

$$h(t,x):={\int }_{{\mathbf{R}}^d}p(t,x;1,x_2){\nu }_2(d{x}_2)\in C^{1,2}([0,1)\times {\mathbf{R}}^d),$$

$$\left(\frac{\partial }{\partial t}+{\mathcal{A}}_t\right)h(t,x)=0,(t,x)\in [0,1)\times {\mathbf{R}}^d.$$

$${\mathcal{A}}_t:=\frac{1}{2}Trace(a(t,x)D_x^2)+⟨b(t,x),D_x⟩.$$

$$dX(t)$$

$$PX(t{)}^{-1}$$

$$PX(t{)}^{-1}(dx)=\left({\int }_{{\mathbf{R}}^d}{\nu }_1(d{x}_1)p(0,x_1;t,x)\right)h(t,x)dx,0\le t\le 1,$$

$${\int }_{{\mathbf{R}}^d}{\nu }_1(d{x}_1)p(0,x_1;0,x)dx:={\nu }_1(dx),$$

$$h(1,x)={\int }_{{\mathbf{R}}^d}{\nu }_2(d{x}_2)p(1,x;1,x_2):=\frac{{\nu }_2(dx)}{dx}.$$

$$h(t,x):=\{\begin{array}{ll}\exp \{u_1(x;p(t,\cdot ;1,\cdot ),PX(t{)}^{-1},P_1)\},& t\in [0,1),\\ \frac{{\nu }_2(dx;p(0,\cdot ;t,\cdot ),P_0,PX(t{)}^{-1})}{dx}& \\ =\exp \{-u_2(x;p(0,\cdot ;t,\cdot ),P_0,PX(t{)}^{-1})\}\frac{PX(t{)}^{-1}(dx)}{dx},& t\in (0,1].\end{array}$$

$$U(t,x,P):=\{\begin{array}{ll}{u}_1(x;p(t,\cdot ;1,\cdot ),P,P_1),t\in [0,1),& \\ & \\ \log \left(\frac{{\nu }_2(dx;p(0,\cdot ;t,\cdot ),P_0,P)}{dx}\right),t=1& \end{array}$$

$${\int }_{{\mathbf{R}}^d}f(x)p(t,x)dx-{\int }_{{\mathbf{R}}^d}f(x)P_0(dx)$$

$$0$$

$$U(1,x,PX(1{)}^{-1})$$

$$d{X}^{\gamma }(t)=\{\gamma (t;\omega )+b(t,X^{\gamma }(t))\}dt+\sigma (t,X^{\gamma }(t))dW(t),$$

$$V(P_0,P_1)$$

$$||\mu||:=\sup\left\{\int_{X}\phi(x)\mu(dx)\biggl{|}\phi\in C(X),||\phi||_{\infty}\leq 1\right\}\in[0,\infty],$$

$$||f|{|}_{\infty }:=\underset{x\in X}{\sup }|f(x)|.$$

$$\underset{n\to \infty }{\lim }(||{\mu }_{1,n}{\mu }_{2,n}-{\mu }_1{\mu }_2||+||q_n-q|{|}_{\infty })=0.$$

Full text

1 Introduction

E. Schrödinger considered a probabilistic problem from which he obtained the so-called Schrödinger’s functional equation (see section 7 in [24] and also [3, 23]). We describe Schrödinger’s functional equation. Let $S$ be a $\sigma$-compact metric space, let $C(S\times S)$ denote the space of all continuous functions on $S\times S$ with the topology induced by the uniform convergence on every compact subset of $S$ and let ${\cal P}(S)$ denote the space of all Borel probability measures on $S$ with the strong topology. Fix a positive function $q\in C(S\times S)$. Schrödinger’s functional equation can be described as follows. For $\mu_{1},\mu_{2}\in{\cal P}(S)$, find a product measure $\nu_{1}(dx_{1})\nu_{2}(dx_{2})$ of nonnegative $\sigma$-finite Borel measures on $S$ for which the following holds:

[TABLE]

It is known that (1.1) has the unique solution (see [6, 12] and also [4, 10]).

[TABLE]

Then $\exp(u_{1}(x))$ and $\exp(u_{2}(x))$ are positive and

[TABLE]

(1.1) can be rewritten as follows: for $i,j=1,2$, $i\neq j$,

[TABLE]

In particular, Schrödinger’s problem (1.1) is equivalent to finding a function $u_{1}(x_{1})+u_{2}(x_{2})$ for which (1.4) holds. Since $\nu_{1}(dx_{1})\nu_{2}(dx_{2})$ is the unique solution of (1.1), it is a functional of $\mu_{1}$, $\mu_{2}$ and $q$. Since it is a product measure, $\nu_{1}$ and $\nu_{2}$ are also functionals of $\mu_{1}$, $\mu_{2}$ and $q$ (see the proof of Corollary 2.1 in section 3):

[TABLE]

This does not imply the uniqueness of $\nu_{1}$ and $\nu_{2}$. Indeed, for $C>0$,

[TABLE]

Let $\{A_{n}\}_{n\geq 1}$ be an nondecreasing sequence of compact subsets of $S$ such that $S=\cup_{n\geq 1}A_{n}$. $A_{1}:=S$ when $S$ is compact. We assume that the following holds so that $\nu_{i}$, $u_{i}$, $i=1,2$ are unique:

[TABLE]

where $n_{0}(\mu_{1},\mu_{2}):=\min\{n\geq 1|\mu_{1}(A_{n})\mu_{2}(A_{n})>0\}.$

Let ${\cal M}(S)$ denote the space of all Radon measures on $S$. In this paper we denote by a Radon measure a locally finite and inner regular Borel measure. It is known that a locally finite and $\sigma$-finite Borel measure on a $\sigma$-compact metric space is a Radon measure in our sense (see e.g., p. 901, Prop. 32.3.4 in [11]).

In Theorem 2.1, we show that if $S$ is compact, then the following are strongly continuous:

[TABLE]

[TABLE]

and $u_{i}\in C(S\times C(S\times S)\times{\cal P}(S)\times{\cal P}(S))$. In Corollary 2.1, we also show that if $S$ is $\sigma$-compact, then the following are weakly Borel measurable and Borel measurable respectively:

[TABLE]

[TABLE]

As an application of this measurability result, we show that the coefficients of the mean field PDE system which the marginal distributions of the h-path process with given two end point marginals satisfy are measurable functions of space, time and marginal. To describe the problem more precisely, we introduce Jamison’s result on SDEs for the h-path process with given two end point marginals. We first describe assumptions and then state Jamison’s results.

(A1.1) $d\geq 1$ and $\sigma(t,x)=(\sigma^{ij}(t,x))_{i,j=1}^{d}$, $(t,x)\in[0,1]\times{\bf R}^{d}$, is a $d\times d$-matrix. $a(t,x):=\sigma(t,x)\sigma(t,x)^{*}$, $(t,x)\in[0,1]\times{\bf R}^{d}$, is uniformly positive definite, bounded, once continuously differentiable and uniformly Hölder continuous. $D_{x}a(t,x)$ is bounded and the first derivatives of $a(t,x)$ are uniformly Hölder continuous with respect to $x$.

(A1.2) $b(t,x):[0,1]\times{\bf R}^{d}\mapsto{\bf R}^{d}$ is bounded, continuous and uniformly Hölder continuous with respect to $x$.

Theorem 1.1

([13], p. 330)* Suppose that (A1.1) and (A1.2) hold. Then for any $P_{0}\in{\cal P}({\bf R}^{d})$, the following SDE has the unique weak solution with a positive continuous transition probability density $p(t,x;s,y)$, $0\leq t<s\leq 1$, $x,y\in{\bf R}^{d}$:*

[TABLE]

Here $W(t)$ denotes a d-dimensional $\sigma[X(s);0\leq s\leq t]$-Brownian motion. Besides, for any $\mu_{1},\mu_{2}\in{\cal P}({\bf R}^{d})$, and the solution $\nu_{2}$ of (1.1) with $S$ and $q(x_{1},x_{2})$ respectively replaced by ${\bf R}^{d}$ and $p(0,x_{1};1,x_{2})$,

[TABLE]

[TABLE]

Here

[TABLE]

Theorem 1.2** (Markovian reciprocal process)**

([13], Theorem 2)* Suppose that (A1.1) and (A1.2) hold. Then for any $P_{0},P_{1}\in{\cal P}({\bf R}^{d})$ for which $P_{1}(dy)\ll dy$, there exists the unique weak solution to the following SDE:*

[TABLE]

Here, to define $h(t,x)$, we consider (1.1) with $\mu_{1},\mu_{2},q(x_{1},x_{2})$ and $S$ respectively replaced by $P_{0},P_{1},p(0,x_{1};1,x_{2})$ and ${\bf R}^{d}$. $W(t)$ also denotes a d-dimensional $\sigma[X(s);0\leq s\leq t]$-Brownian motion. Besides,

[TABLE]

where

[TABLE]

[TABLE]

Remark 1.1. Replace $S$ by ${\bf R}^{d}$ in (1.1). Then the following holds (see (1.2), (1.3), (1.8) and (1.11)): for $x\in{\bf R}^{d}$,

[TABLE]

As an application of Corollary 2.1 in section 2, we show that

[TABLE]

is a Borel measurable function from $[0,1]\times{\bf R}^{d}\times{\cal P}({\bf R}^{d})$ to ${\bf R}$ (see Corollary 2.2). Theorems 1.1 and 1.2 and (1.12)-(1.13) imply that if $P_{1}(dy)\ll dy$, then $p(t,x)dx:=PX(t)^{-1}(dx)$ satisfies the following mean field PDE system (see [1, 2, 5, 14] and the references therein for the mean field games and the master equations). For any $f\in C_{b}^{2}({\bf R}^{d})$ and $t\in(0,1]$,

[TABLE]

and for $(t,x)\in(0,1)\times{\bf R}^{d}$,

[TABLE]

[TABLE]

Here we consider $U(t,x,PX(t)^{-1})$ as a function of $(t,x)$.

Let $\gamma(t;\omega)$ denote a progressively measurable ${\bf R}^{d}$-valued stochastic process on some filtered probability space and consider the following SDE in a weak sense:

[TABLE]

provided it exists (see e.g. [8]). Here $W(t)$ denotes a $d$-dimensional Brownian motion defined on the same filtered probability space as $\gamma(t;\omega)$.

It is also known that the h-path process with given two end point marginals is the unique minimizer of the following stochastic optimal control problem (see [7, 9], [15]-[22], [25], [26] and the references therein for recent progress, especially for stochastic optimal transport).

Theorem 1.3

([7], [21], [26])* Suppose that (A1.1) and (A1.2) hold. Then for any $P_{0},P_{1}\in{\cal P}({\bf R}^{d})$ for which $P_{1}(dy)\ll dy$, $\gamma(t;\omega)=a(t,X^{\gamma}(t))D_{x}\log h(t,X^{\gamma}(t))$ is the unique minimizer of the following:*

[TABLE]

provided it is finite (see (1.10) for notation).

Remark 1.2. A sufficient condition for the finiteness of $V(P_{0},P_{1})$ is given in [20] for more general problems.

Schrödinger’s functional equation (1.1) with $q(x_{1},x_{2})$ and $S$ respectively replaced by $p(0,x_{1};1,x_{2})$ and ${\bf R}^{d}$ is equivalent to the Euler equation for $V(P_{0},P_{1})$. We state and prove it for readers’ convenience since we could not find any literature (see Proposition 2.1).

In section 2 we state our main results and prove them in section 3.

2 Main results

In this section we state our main results. We first describe assumptions.

(A2.1) $S$ is a compact metric space.

(A2.2) $q\in C(S\times S;(0,\infty))$.

(A2.1)’ $S$ is a $\sigma$-compact metric space.

For a metric space $X$ and $\mu\in{\cal M}(X)$,

[TABLE]

where for $f\in C(X)$,

[TABLE]

When $S$ is compact, we have the continuity results on $\nu_{i}$, $u_{i}$ in (1.5) (Recall (1.6)).

Theorem 2.1

Suppose that (A2.1) and (A2.2) hold. Suppose also that $\mu_{i,n}$, $\mu_{i}\in{\cal P}(S)$, $q_{n}\in C(S\times S;(0,\infty))$, $i=1,2$, $n\geq 1$ and

[TABLE]

*Then *

[TABLE]

Besides, for $i=1,2$, and $\{x_{n}\}_{n\geq 1}\subset S$ which converges, as $n\to\infty$, to $x\in S$,

[TABLE]

When $S$ is $\sigma$-compact, we only have the Borel measurability results on $\nu_{i}$, $u_{i}$ in (1.5).

Corollary 2.1

Suppose that (A2.1)’ and (A2.2) hold. Then the following are Borel measurable: for $i=1,2$,

[TABLE]

[TABLE]

As an application of Corollary 2.1, we obtain the following.

Corollary 2.2

Suppose that (A1.1) and (A1.2) hold. Then $U(t,x,P)$ in (1.13) is a Borel measurable function from $[0,1]\times{\bf R}^{d}\times{\cal P}({\bf R}^{d})$ to ${\bf R}$. In particular, (1)-(1) hold.

For $P_{0}\in{\cal P}({\bf R}^{d})$ and Borel measurable $f:{\bf R}^{d}\mapsto{\bf R}$,

[TABLE]

(see (1.3) for notation). Then since $P\mapsto V(P_{0},P)$ is convex, lower semicontinuous and $\not\equiv\infty$, for $P\in{\cal P}({\bf R}^{d})$,

[TABLE]

(see [18, 19, 21, 25] and the references therein). The following gives the variational meaning to Schrödinger’s functional equation.

Proposition 2.1

Suppose that (A1.1) and (A1.2) hold. Then for any $P_{0},P_{1}\in{\cal P}({\bf R}^{d})$ for which $P_{1}(dy)\ll dy$ and for which $V(P_{0},P_{1})$ is finite, Schrödinger’s functional equation (1.1) with $\mu_{1}$, $\mu_{2}$ and $q(x_{1},x_{2})$ respectively replaced by $P_{0}$, $P_{1}$ and $p(0,x_{1};1,x_{2})$ is equivalent to the following:

[TABLE]

Here $\displaystyle\frac{\delta V_{P_{0}}^{*}(f)}{\delta f}$ denotes the Gâteaux derivative of $V_{P_{0}}^{*}(f)$.

3 Proof of main results

In this section we state and prove lemmas and prove our main results.

[TABLE]

The following two lemmas are proved in [4].

Lemma 3.1

([4], p. 194)* Suppose that (A2.1) and (A2.2) hold. Then, for any $\mu_{1},\mu_{2}\in{\cal P}(S)$, there exists a unique pair of nonnegative finite measures $\nu_{1}$, $\nu_{2}$ on $S$ for which (1.1) and the following holds:*

[TABLE]

[TABLE]

(see (1.2) for notation).

Lemma 3.2

([4], section 7)* Suppose that (A2.1) and (A2.2) hold. Then, there exists a function $c(a,b)$ which is nonincreasing in $a$ and nondecreasing in $b$ such that for any $\mu_{i},\tilde{\mu}_{i}\in{\cal P}(S)$, $i=1,2$,*

[TABLE]

Here $\tilde{\nu}_{i}(dx):=\nu_{i}(dx;q,\tilde{\mu}_{1},\tilde{\mu}_{2})$, $i=1,2$ (see (1.5) and (2.1) for notation).

The following lemma can be proved by Lemma 3.1.

Lemma 3.3

Suppose that (A2.1) and (A2.2) hold and that $q_{n}\in C(S\times S;(0,\infty))$, $n\geq 1$ and

[TABLE]

(see (2.2) for notation). Then, for any $\mu_{i}\in{\cal P}(S)$, $i=1,2$,

[TABLE]

where $\nu_{n,i}(dx):=\nu_{i}(dx;q_{n},\mu_{1},\mu_{2})$ (see (1.5) and (2.1) for notation).

(Proof) $u_{n,i}(x):=u_{i}(x;q_{n},\mu_{1},\mu_{2})$. Then, from (1.2)-(1.3),

[TABLE]

For $i=1,2$, $\displaystyle\left\{\frac{\nu_{n,i}(dx)}{\nu_{n,i}(S)}\right\}_{n\geq 1}$ is a tight family of probability measures and $\{\nu_{n,i}(S)\}_{n\geq 1}$ is bounded from above and below by (3.2). In particular, there exist $\{s(n)\}_{n\geq 1}$ and a finite measure $\overline{\nu}_{i}$ such that $\nu_{s(n),i}$ weakly converges, as $n\to\infty$, to $\overline{\nu}_{i}$. From construction, (3.2) with $\nu_{i}$ replaced by $\overline{\nu}_{i}$ also holds.

[TABLE]

Then for $i=1,2$,

[TABLE]

Indeed, from (3.7),

[TABLE]

For $i,j=1,2,i\neq j$ and $x_{i}\in S$,

[TABLE]

from (3.2) and (3.5). From (3.3),

[TABLE]

In particular, the bounded convergence theorem implies that (3.9) is true.

From (3.8)-(3.9),

[TABLE]

The uniqueness of the solution to (1.1) implies that

[TABLE]

since (3.2) hold for both of $\overline{\nu}_{i}$ and $\nu_{i}$. Since the above method applies for any subsequence of $\{q_{n}\}_{n\geq 1}$, the discussion in (3.9) implies that the following holds:

[TABLE]

(3.2) and (3.12) completes the proof.$\Box$

We prove Theorem 2.1 by Lemmas 3.1-3.3.

(Proof of Theorem 2.1)

Lemmas 3.2 and 3.3 imply (2.4). We prove (2.5). Without loss of generality, we only have to consider the case when $i=1$. For sufficiently large $n\geq 1$,

[TABLE]

We prove (3.13). For $x\in S$,

[TABLE]

[TABLE]

from(3.2)-(3.3). The following also holds:

[TABLE]

Indeed, for $f\in C_{b}(S)$ for which $||f||_{\infty}\leq 1$, from (2.4) and (3.2),

[TABLE]

The following completes the proof of (3.13):

[TABLE]

We prove (2.6). From (2.5), we only have to prove the following: for $i=1,2$, and $\{x_{n}\}_{n\geq 1}\subset S$ which converges to $x\in S$ as $n\to\infty$,

[TABLE]

This can be proved by the bounded convergence theorem.$\Box$

For $\mu_{1},\mu_{2}\in{\cal P}(S)$,

[TABLE]

where ${\cal B}(S)$ denotes the Borel $\sigma$-field of $S$ (see (1.6) for notation). When we replace $X$ and $S$ by $A_{n}$ in (2.1)-(2.2) and (3.1), we use notations $||\cdot||_{n}$, $||\cdot||_{\infty,n}$, $m_{q,n}$ and $M_{q,n}$ instead of $||\cdot||$, $||\cdot||_{\infty}$, $m_{q}$ and $M_{q}$ respectively. We use a similar convention when it is not confusing.

We introduce and prove two lemmas to prove Corollary 2.1.

Lemma 3.4

Suppose that (A2.1)’ and (A2.2) hold. Then, for any $\mu_{1},\mu_{2}\in{\cal P}(S)$ and any $k\geq n_{0}(\mu_{1},\mu_{2})$, there exists a unique pair of nonnegative finite measures $\nu_{1|k}$, $\nu_{2|k}$ on $A_{k}$ for which Lemma 3.1 with $S$, $m_{q}$, $M_{q}$, $\mu_{i}$, $\nu_{i}$, $u_{i}$, $i=1,2$ replaced by $A_{k}$, $m_{q,k}$, $M_{q,k}$, $\mu_{i|k}$, $\nu_{i|k}$, $u_{i|k}$, $i=1,2$ respectively holds. Suppose, in addition, that $\mu_{i,n}\in{\cal P}(S)$, $q_{n}\in C(S\times S;(0,\infty))$, $i=1,2$, $n\geq 1$ and

[TABLE]

Then (2.4)-(2.6) hold even if $\nu_{i}$, $\mu_{i,n}$, $\mu_{i}$, $||\cdot||$, $u_{i}$, $||\cdot||_{\infty}$ and $S$ is replaced by $\nu_{i|k}$, $\mu_{i,n|k}$, $\mu_{i|k}$, $||\cdot||_{k}$, $u_{i|k}$, $||\cdot||_{\infty,k}$ and $A_{k}$ respectively.

(Proof) Theorem 2.1 and the following completes the proof:

[TABLE]

(3.16) is true, since

[TABLE]

For any $\mu_{1},\mu_{2}\in{\cal P}(S)$ and any $n\geq n_{0}(\mu_{1},\mu_{2})$,

[TABLE]

The following is known.

Lemma 3.5

([12], Theorem 3.2)* Suppose that (A2.1)’ and (A2.2) hold. Then for any $\mu_{i}\in{\cal P}(S)$, $i=1,2$, there exists a unique solution $\nu_{1}(dx_{1})\nu_{2}(dx_{2})$ to (1.1) and $\mu^{(n)}(dx_{1}dx_{2})$ weakly converges, as $n\to\infty$, to $\mu(dx_{1}dx_{2}):=q(x_{1},x_{2})\nu_{1}(dx_{1})\nu_{2}(dx_{2})$.*

By Lemmas 3.4 and 3.5, we prove Corollary 2.1.

(Proof of Corollary 2.1) Without loss of generality, we only have to prove the case when $i=1$. From Lemma 3.4, for any $n\geq 1$, $\int_{S\times S}f(x_{1},x_{2})\mu^{(n)}(dx_{1}dx_{2})$ is continuous in $(f,q,\mu_{1},\mu_{2})$ on the open set

[TABLE]

(see (1.6)). Notice that $n_{0}(\mu_{1},\mu_{2})\leq n$ if and only if $\mu_{1}(A_{n})\mu_{2}(A_{n})>0$. From Lemma 3.5, $\int_{S\times S}f(x_{1},x_{2})\mu(dx_{1}dx_{2})$ is measurable in $(f,q,\mu_{1},\mu_{2})$. The following implies the first part of Corollary 2.1: for $f\in C_{o}(S)$ and $x\in\mathbb{R}$,

[TABLE]

For any $x\in S$ and $\phi_{n}\in C_{o}(S)$ for which $0\leq\phi_{n}\leq 1$ and $\phi_{n}(y)=1,y\in A_{n}$, $\int_{S}\phi_{n}(x_{2})q(x,x_{2})\nu_{2}(dx_{2})$ is measurable in $(q,\mu_{1},\mu_{2})$ in the same way as above and is continuous in $x$. In particular, it is measurable in $(x,q,\mu_{1},\mu_{2})$ and so is the following: by Fatou’s lemma,

[TABLE]

Corollary 2.1 immediately implies Corollary 2.2.

(Proof of Corollary 2.2) Since $p(t,\cdot;1,\cdot)$ is continuous on $[0,1)$ from Theorem 1.1,

[TABLE]

is continuous on $[0,1)\times{\bf R}^{d}\times{\cal P}({\bf R}^{d})\times{\cal P}({\bf R}^{d})$, which implies the measurability of $U(t,x,P)$. It is easy to see that (1.14) - (1.15) hold.$\Box$

We prove Proposition 2.1.

(Proof of Proposition 2.1)

[TABLE]

Indeed, for any $\psi\in C_{b}({\bf R}^{d})$ and $\varepsilon\in{\bf R}$, instead of $P_{1}$, consider Schrödinger’s problem (1.1) with $S,\mu_{1},\mu_{2}$ and $q(x_{1},x_{2})$ respectively replaced by ${\bf R}^{d},P_{0},\mu^{\varepsilon\psi}$ and $q(0,x_{1};1,x_{2})$, where for $f\in C_{b}({\bf R}^{d})$

[TABLE]

(see (1.1)-(1.4)). Then, from Theorem 1.3 and (2.8) (see e.g. [7, 26] and also [21]),

[TABLE]

(see (1.8)). This implies (3.18). From (1.8),

[TABLE]

(3.18) and (3.19) completes the proof.$\Box$