# Optimal transport in the frame of abstract Lax-Oleinik operator revisited

Wei Cheng, Jiahui Hong and Tianqi Shi

###### Abstract.

This is our first paper on the extension of our recent work on the Lax-Oleinik
commutators and its applications to the intrinsic approach of propagation of
singularities of the viscosity solutions of Hamilton-Jacobi equations. We
reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal
transport in terms of abstract Lax-Oleinik operators, and analyze the relevant
optimal transport problem in the case the cost function
$c(x,y)=h(t_{1},t_{2},x,y)$ is the fundamental solution of Hamilton-Jacobi
equation. For further applications to the problem of cut locus and propagation
of singularities in optimal transport, we introduce corresponding random Lax-
Oleinik operators. We also study the problem of singularities for $c$-concave
functions and its dynamical implication when $c$ is the fundamental solution
with $t_{2}-t_{1}\ll 1$ and $t_{2}-t_{1}<\infty$, and $c$ is the Peierls’
barrier respectively.

##  1\. Introduction

This is the first paper of further extension of our recent work on the Lax-
Oleinik commutators and its applications to the problem of propagation of
singularities of the viscosity solutions of Hamilton-Jacobi equations [17].
The relation of optimal transport and weak KAM theory was first found by
Bernard and Buffoni in the papers [8, 9, 10]. The readers can refer to [27]
for a systematic treatment of the optimal transport problem for the cost of
Lagrangian action. For the squared distance cost function and the associated
Hopf-Lax semigroup, see, for instance [5, 3]. The purpose of this paper is to
revisit the optimal transport problem for Lagrangian action. It is motivated
by our recent work on Lax-Oleinik commutators and its applications to the
problem of cut locus and singularities in optimal transport which is not
essentially well understood before. The applications of the main setting and
results to this topic will be in our current papers in preparation.

###  1.1. Singularities of $c$-concave functions

Let $X$ and $Y$ be two Polish spaces and $c:X\times Y\to\mathbb{R}$. The
function $c$ is called _cost function_. We consider an abstract setting of
Lax-Oleinik operators. For any $\phi:X\to\mathbb{R}$ and $\psi:Y\to\mathbb{R}$
we define the _abstract Lax-Oleinik operators_

$\displaystyle T^{-}\phi(y):=\inf_{x\in X}\\{\phi(x)+c(x,y)\\},\qquad
T^{+}\psi(x):=\sup_{y\in Y}\\{\psi(y)-c(x,y)\\},\qquad x\in X,y\in Y.$

Recall that a function $\psi:Y\to[-\infty,+\infty)$ is said to be _$c$
-concave_ if it is the infimum of a family of functions
$c(x,\cdot)+\alpha(x)$. Analogously, $\phi:X\to(-\infty,+\infty]$ is said to
be _$c$ -convex_ if it is the supreme of a family of functions
$\beta(y)-c(\cdot,y)$. Given $\psi:Y\to[-\infty,+\infty)$ and
$\psi(y)>-\infty$ we define the $c$-superdifferential as

$\displaystyle\partial^{c}\psi(y):=\\{x\in X:\psi(\cdot)-c(x,\cdot)\
\text{attains maximum at}\ y\\}.$

For any $c$-concave function $\psi:Y\to\mathbb{R}$, $y\in Y$ is called a
_singular point_ of $\psi$ if $\partial^{c}\psi(y)$ is not a singleton. For
any $c$-concave function $\psi:Y\to\mathbb{R}$, we denote by
$\text{Sing}^{c}(\psi)$ the set of all singular points of $\psi$.

By using this setting we first obtain the characterization of $c$-concavity
via Lax-Oleinik commutators: the following statements are equivalent (see
Theorem 3.6).

  1. (1)

$T^{-}\circ T^{+}\psi=\psi$.

  2. (2)

$\partial^{c}\psi(y)\not=\varnothing$ for all $y\in Y$, where
$\partial^{c}\psi(y):=\\{x\in X:\psi(\cdot)-c(x,\cdot)\ \text{attains maximum
at}\ y\\}$.

  3. (3)

There exists $\phi:X\to\mathbb{R}$ such that $\psi=T^{-}\phi$.

  4. (4)

$\psi$ is $c$-concave.

The problem of singularities of semiconcave functions plays an important role
in calculus of variation and optimal control, PDE, Hamiltonian dynamical
systems and geometry. For convex Lagrangian on Euclidean space or finite
dimensional manifold, the maximal regularity of the associated value function
is semiconcavity. More precisely, for any Tonelli Lagrangian
$L:\mathbb{R}\times TM\to\mathbb{R}$ with $M$ an $d$-dimensional smooth
manifold and a function $\phi:M\to\mathbb{R}$, let

$\displaystyle
u(t,x):=\inf_{\gamma}\left\\{\phi(\gamma(0))+\int^{t}_{0}L(s,\gamma(s),\dot{\gamma}(s))\
ds\right\\},\qquad t>0,x\in M,$

where $\gamma:[0,t]\to M$ is taken over the set of absolutely continuous
curves such that $\gamma(t)=x$. Then the function $u$ is locally semiconcave
on $(0,+\infty)\times M$. We say a point $(t,x)\in(0,+\infty)\times M$ a
singular point of $u$ if $u$ is not differentiable at $(t,x)$. We denote by
$\text{Sing}\,(u)$ the set of all singular points of $u$. If $u$ is only a
locally semiconcave function on $M$, we also denote the singular set of $u$ by
$\text{Sing}\,(u)$.

There is a large amount of literatures on the structure of $\text{Sing}\,(u)$
from topological, measure theoretic and even to dynamical aspects. For a
survey the readers can refer to [14]. The rectifiability result of
$\text{Sing}\,(u)$ plays an important role in the solvability issue of Monge
problem in optimal transport problem, for cost function which is the squared
distance on Euclidean space or Riemannian manifold. However, for general cost
functions, the relation between $\text{Sing}^{c}(\psi)$ and the solution of
relevant Monge problem is much more complicated. In [7], the authors discuss
this problem under rather general conditions. However, invoking our recent
work on the Lax-Oleinik commutators with applications to the singularities in
the context of weak KAM theory, we realize it is useful to understand
$\text{Sing}^{c}(\psi)$ and their dynamical nature.

In the context of weak KAM theory, we suppose $X=Y=M$ with $M$ a compact and
connected manifold without boundary. Given any $x,y\in M$, and $t_{1}<t_{2}$,
we denote by $\Gamma_{x,y}^{t_{1},t_{2}}$ the set of absolutely continuous
curves $\gamma\in\text{AC}([t_{1},t_{2}],M)$ with $\gamma(t_{1})=x$ and
$\gamma(t_{2})=y$. Recall that

$\displaystyle
h(t_{1},t_{2},x,y):=\inf_{\gamma\in\Gamma_{x,y}^{t_{1},t_{2}}}\int_{t_{1}}^{t_{2}}L(s,\gamma(s),\dot{\gamma}(s))\,ds,\qquad
t_{1}<t_{2},\quad x,y\in M,$

and $h(x,y)=\liminf_{t_{2}-t_{1}\to\infty}h(t_{1},t_{2},x,y)$ is the Peierls’
barrier with a time-independent Lagrangian $L$ and the Mañé’s critical value
is $0$.

To study the singularities of $c$-concave functions, we will consider three
types of cost functions in different cases: $c(x,y)=h(t_{1},t_{2},x,y)$ with
$t_{2}-t_{1}\ll 1$, $c(x,y)=h(t_{1},t_{2},x,y)$ with finite $t_{1}<t_{2}$ and
$c(x,y)=h(x,y)$.

A key observation for the first case is due to a theorem by Marie-Claude
Arnaud ([6]) on the evolution of the 1-graph of semiconcave functions under
Hamiltonian flow for short time. In this case, we conclude $\text{\rm
Sing}^{c}(\psi)=\text{\rm Sing}\,(\psi)$ (Proposition 3.12). In the case with
finite $t_{1}<t_{2}$, one can only have the inclusion
$\text{Sing}^{c}(\psi)\subset\text{Sing}\,(\psi)$ (Proposition 3.14). For the
case $c(x,y)=h(x,y)$ and $u^{-}$ is a weak KAM solution, we prove for any
$y\in M$, there exists a static class
$\mathcal{A}_{y}\subset\partial^{c}u^{-}(y)$ (Proposition 3.15). Recall that
the static class is the equivalence class determined by the pseudo-distance
$d(x,y)=h(x,y)+h(y,x)$ on the Aubry set $\mathcal{A}$ ([25, 21]). As a
consequence, if each static class is not a singleton, then $\mbox{\rm
Sing}^{c}(u^{-})=M$.

###  1.2. An alternative formulation of Kantorovich-Rubinstein duality

Let $\phi:X\to\mathbb{R}$ and $\psi:Y\to\mathbb{R}$. Set
$c_{\phi}(x,y):=\phi(x)+c(x,y)$ and $c^{\psi}(x,y):=\psi(y)-c(x,y)$. For any
$\mu\in\mathscr{P}(X)$ and $\nu\in\mathscr{P}(Y)$, we try to find a function
$\phi:X\to\mathbb{R}$ and a function $\psi:Y\to\mathbb{R}$ such that

$\displaystyle\int_{Y}T^{-}\phi\ d\nu=$
$\displaystyle\,\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times Y}c_{\phi}(x,y)\
d\pi,$ (K-) $\displaystyle\int_{X}T^{+}\psi\ d\mu=$
$\displaystyle\,\sup_{\pi\in\Gamma(\mu,\nu)}\int_{X\times Y}c^{\psi}(x,y)\
d\pi.$ (K+)

In terms on abstract Lax-Oleinik operators, we also proved the equivalence of
the problems (K-) and (K+) and well known Rubinstein-Kantorovich duality (see
Theorem 4.2). More precisely,

  1. –

if $\phi:X\to\mathbb{R}$ is a solution of (K-), then $(\phi,T^{-}\phi)$ is a
solution of Kantorovich problem. Conversely, if $(\phi,\psi)\in I_{c}$ is a
solution of Kantorovich problem, then $\phi$ solves (K-);

  2. –

if $\psi:Y\to\mathbb{R}$ is a solution of (K+), then $(T^{+}\psi,\psi)$ is a
solution of Kantorovich problem. Conversely, if $(\phi,\psi)\in I_{c}$ is a
solution of Kantorovich problem, then $\psi$ solves (K+),

where

$\displaystyle I_{c}:=\\{(\phi,\psi):\psi(y)-\phi(x)\leqslant c(x,y)\
\text{for all}\ x\in X,y\in Y\\}.$

Invoking the equivalence above between problem (K-) (or (K+)) and the
Rubinstein-Kantorovich duality, it is useful to refine the analysis of the
dynamical nature of the associated optimal transport problem initiated by
Bernard and Buffoni ([8, 9, 10]).

To study the singularities in the optimal transport, we have to introduce some
“Random” Lax-Oleinik operators (Definition 4.21). However, if $X=Y=M$, a
connected closed manifold, $c$ is the fundamental solution of the associated
Hamilton-Jacobi equation and the terminal marginal measure $\nu$ is Dirac. The
usual deterministic optimal transport problem can be understood well in terms
of the associated Hamiltonian flow. Our main result (Proposition 4.7) in this
case is: let $y_{0}\in M$, $\nu=\delta_{y_{0}}$, $\phi\in\text{Lip}_{b}(M)$
and $\\{\Phi_{H}^{t_{1},t_{2}}\\}_{t_{1},t_{2}\in\mathbb{R}}$ is the
associated Hamiltonian flow from $t_{1}$ to $t_{2}$.

  1. (1)

For any $\rho\in\mathscr{P}(\mathbb{R}^{d})$ with
$\mathrm{supp}\,(\rho)\subset D^{*}T^{-}\phi(y_{0})$111For a semiconcave
function $u$, $D^{*}u(x)$ is the set of reachable gradients., there exists
$\mu_{\rho}:=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho)$,
which makes (K-) holds true;

  2. (2)

For each $\mu\in\mathscr{P}(M)$ which satisfies (K-), there exists
$\rho_{\mu}\in\mathscr{P}(\mathbb{R}^{d})$ with
$\mathrm{supp}\,(\rho_{\mu})\subset D^{+}T^{-}\phi(y_{0})$, such that
$\mu=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho_{\mu})$.

In the case that $\nu$ is a general probability measure, we also have the
following result. Let $\nu\in\mathscr{P}(M)$ and $\phi\in\text{Lip}\,(M)$.
Then, there exists some $\mu\in\mathscr{P}(M)$ solving (K-) and satisfying

$\displaystyle\mathrm{supp}\,(\mu)\subset\mathop{\lim\inf}_{k\to\infty}\left\\{x\in
M:\partial_{c}\phi(x)\cap C_{k}\neq\varnothing\right\\},$

for some $a_{k}$-net $C_{k}$ of $\mathrm{supp}\,\,(\nu)$, where $a_{k}\searrow
0$ as $k\to\infty$.

###  1.3. Random Lax-Oleinik operators

The introduction of random Lax-Oleinik operators is a key step for the problem
of cut locus and propagation of singularities in optimal transport. We use the
word “random” to clarify the essential difference of the stochastic setting of
this problem, where sample paths of the relevant processes is not absolutely
continuos.

For a metric space $(X,d)$, let $(\mathscr{P}_{p}(X),W_{p})$ be the
$p$-Wasserstein space for $p\in[1,+\infty)$, where $W_{p}$ is the $p$-order
Wasserstein metric. Let $s,t\in\mathbb{R}$ with $s<t$, $\mu\in\mathscr{P}(X)$
and $\nu\in\mathscr{P}(Y)$. We follow the setting of [27]. The dynamical cost
functional associated to $c^{s,t}$ is

$\displaystyle C^{s,t}(\mu,\nu):=\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times
Y}c^{s,t}(x,y)\,d\pi=\inf_{\begin{subarray}{c}\mathrm{law}(x)=\mu\\\
\mathrm{law}(y)=\nu\end{subarray}}\mathbb{E}(c^{s,t}(x,y)),$

where $x(\omega)$ and $y(\omega)$ in the last term are $X$-valued and
$Y$-valued random variables of some probability space
$(\Omega,\mathscr{F},\mathbb{P})$ respectively.

Given $\phi\in L^{0}(X;(-\infty,+\infty])$ which is bounded from below,
associated with $\phi$ we define a functional of $\mathscr{P}_{p}(X)$, called
the potential energy associated to $\phi$ (see [4, Definition 4.30] or [5,
Example 9.3.1]),

$\displaystyle\phi(\mu):=\int_{X}\phi\,d\mu,\qquad\mu\in\mathscr{P}_{p}(X).$

Let $X=Y=M$ and $c^{s,t}(x,y):=h(s,t,x,y)$, the fundamental solution from weak
KAM theory. For any function $\phi:\mathbb{R}^{d}\to[-\infty,+\infty]$, $t>0$,
$\mu\in\mathscr{P}_{1}(\mathbb{R}^{d})$, we define

$\displaystyle P_{t}^{-}\phi(\mu)$
$\displaystyle:=\inf_{\nu\in\mathscr{P}_{1}(\mathbb{R}^{d})}\\{\phi(\nu)+C^{0,t}(\nu,\mu)\\},$
$\displaystyle P_{t}^{+}\phi(\mu)$
$\displaystyle:=\sup_{\nu\in\mathscr{P}_{1}(\mathbb{R}^{d})}\\{\phi(\nu)-C^{0,t}(\mu,\nu)\\}.$

We call $P_{t}^{\pm}\phi$ the positive and negative _random Lax-Oleinik
operator_.

Our main result is: suppose $\phi$ is a uniformly continuous function on $M$,
$t>0$. Then, for any $\nu\in\mathscr{P}_{1}(M)$ the operator
$P_{t}^{-}\phi(\nu)$ is finite-valued, there exist $\mu\in\mathscr{P}_{1}(M)$
and $\xi\in L_{\mu,\nu}^{0,t}$ such that

$\displaystyle P_{t}^{-}\phi(\nu)$ $\displaystyle=\phi(\mu)+C^{0,t}(\mu,\nu)$
$\displaystyle=\int_{\Omega}\phi(\xi(0,\omega))+\int_{0}^{t}L(\xi(s,\omega),\dot{\xi}(s,\omega))\,dsd\mathbb{P}$
$\displaystyle=\int_{0}^{t}\int_{T\mathbb{R}^{d}}\phi(x)+L(x,v)\,d\tilde{\mu}_{s}ds,$

where $\tilde{\mu}_{s}:=\mathrm{law\,}(\xi(s,\cdot),\dot{\xi}(s,\cdot))$ is
determined by the associated Euler-Lagrangian flow. Moreover,
$P_{t}^{-}\phi(\nu)=T_{t}^{-}\phi(\nu)$ for any $\nu\in\mathscr{P}_{1}(M)$,
i.e., for any $\nu\in\mathscr{P}_{1}(M)$, there exists
$\mu\in\mathscr{P}_{1}(M)$ solving (K-).

We remark finally if the Lagrangian $L$ is a Riemannian metric, the associated
“Random” Hopf-Lax semigroup has already been widely used in the community of
optimal transport (see, for instance, [3]). However, we emphasize the abstract
and random Lax-Oleinik operators for general Lagrangian is useful for our
program to deal with the problem of propagation of singularities based on our
intrinsic approach of this problem in the context of calculus of variation
([15, 13, 16]).

The paper is organized as follows. In section 2, we introduce some necessary
facts from weak KAM theory, calculus of variation and optimal transport. In
Section 3, we define the abstract negative and positive Lax-Oleinik operators
$T^{\pm}$ and characterize the conditions $T^{-}\circ T^{+}=Id$ and
$T^{+}\circ T^{-}=Id$. We also discuss the relations between the singular set
of semiconcave functions and $c$-concave function in the context of weak KAM
theory when $c$ is the fundamental solutions and Peierls’ barrier. Section 4
is composed of three parts. We obtain an equivalent form of the Kantorovich-
Rubinstein duality using the Lax-Oleinik operators $T^{\pm}$. We also discuss
some basic problems of optimal transport in the frame of Lax-Oleinik operators
$T^{\pm}$. In the last subsection, we introduce the random Lax-Oleinik
operators and discuss some fundamental properties of these operators in frame
of abstract Lax-Oleinik operators. We give the proof of a possible known lemma
in the appendix.

Acknowledgements. Wei Cheng is partly supported by National Natural Science
Foundation of China (Grant No. 12231010). Jiahui Hong is partly supported by
Super Postdoctoral Incentive Plan of Shanghai “Singularity problems of
Hamilton-Jacobi equations”.

##  2\. Preliminaries

###  2.1. Semiconcave functions

We first recall some basic relevant notions on semiconcavity.

  1. –

Let $\Omega$ be an open convex subset of $\mathbb{R}^{d}$. A function
$\phi:\Omega\to\mathbb{R}$ is called a semiconcave function (of linear
modulus) with constant $C\geqslant 0$ if for any $x\in\Omega$ there exists
$p\in\mathbb{R}^{d}$ such that

$\phi(y)\leqslant\phi(x)+\langle
p,y-x\rangle+\frac{C}{2}|x-y|^{2},\qquad\forall y\in\mathbb{R}^{d}.$ (2.1)

  2. –

The set of covectors $p$ satisfying (2.1) is called the proximal
superdifferential of $\phi$ at $x$ and we denote it by $D^{+}\phi(x)$.

  3. –

Similarly, $\phi$ is semiconvex if $-\phi$ is semiconcave. The set of
$D^{-}\phi(x)=-D^{+}(-\phi)(x)$ is called the proximal subdifferential of
$\phi$ at $x$.

  4. –

The set $D^{+}\phi(x)$ is a singleton if and only if $\phi$ is differentiable
at $x$, and $D^{+}\phi(x)=\\{D\phi(x)\\}$. A point $x$ is called a singular
point of a semiconcave function $\phi$ if $D^{+}\phi(x)$ is not a singleton.
We denote by $\text{Sing}\,(\phi)$ the set of all singular points of $\phi$.

  5. –

We call $p\in D^{*}\phi(x)$, the set of reachable gradients, if there exists a
sequence $x_{k}\to x$ as $k\to\infty$, $\phi$ is differentiable at each
$x_{k}$ and $p=\lim_{k\to\infty}D\phi(x_{k})$. We have $D^{*}\phi(x)\subset
D^{+}\phi(x)$ and $D^{+}\phi(x)=\text{co}\,D^{*}\phi(x)$.

For more on the semiconcave functions, the readers can refer to [19].

######  Proposition 2.1.

A function $\phi:\Omega\to\mathbb{R}$ is a semiconcave function with constant
$C\geqslant 0$ if and only if there exists a family of $C^{2}$-functions
$\\{\phi_{i}\\}$ with $D^{2}\phi_{i}\leqslant CI$ such that

$\displaystyle\phi=\inf_{i}\phi_{i}.$

Proposition 2.1 is an important and useful characterization of semiconcavity.
Let $S$ be a compact topological space and
$F:S\times\mathbb{R}^{d}\to\mathbb{R}$ be a continuous function such that

  1. (a)

$F(s,\cdot)$ is of class $C^{2}$ for all $s\in S$ and $\|F(s,\cdot)\|_{C^{2}}$
is uniformly bounded by some constant $C$,

  2. (b)

$D_{x}F(s,x)$ is continuous on $S\times\mathbb{R}^{d}$,

  3. (c)

$\phi(x)=\inf\\{F(s,x):s\in S\\}$.

We call such a function $\phi$ a marginal function of a family of
$C^{2}$-functions.

######  Proposition 2.2.

Let $\phi(x)=\inf\\{F(s,x):s\in S\\}$ be a marginal functions of the family of
$C^{2}$-functions $\\{F(s,\cdot)\\}_{s\in S}$. Let
$M(x)=\arg\min\\{F(s,x):s\in S\\}$.

  1. (1)

$\phi$ is semiconcave with constant $C$.

  2. (2)

$D^{*}\phi(x)\subset Y(x)=\\{D_{x}F(s,x):s\in M(x)\\}$.

  3. (3)

For each $x$

$\displaystyle D^{+}\phi(x)=\begin{cases}D_{x}F(s,x),&M(x)=\\{s\\}\ \text{is a
singleton};\\\ \mbox{\rm co}\,\\{D_{x}F(s,x):s\in
M(x)\\},&\text{otherwise.}\end{cases}$

######  Remark 2.3.

If $M$ is a connected and compact smooth manifold without boundary, a function
$\phi:M\to\mathbb{R}$ is called semiconcave if there exists a family of
$C^{2}$-functions $\\{\phi_{i}\\}$ such that $\phi=\inf_{i}\phi_{i}$, and the
Hessians of $\phi_{i}$’s are uniformly bounded above. The readers can refer to
the appendix of the paper [23] for more discussion of the semiconcavity of
functions on (even noncompact) manifold. However, because of the local nature
of our main discussion, we will work on Euclidean space instead.

###  2.2. Fundamental solutions and Lax-Oleinik representation

A function $L=L(t,x,v):\mathbb{R}\times TM\to\mathbb{R}$ is called time-
dependent _Tonelli Lagrangian_ , if it is a function of class $C^{2}$ and
satisfies the following conditions:

  1. (L1)

The function $v\mapsto L(t,x,v)$ is strictly convex for all
$(t,x)\in\mathbb{R}\times M$.

  2. (L2)

There exist a superlinear function $\theta:[0,+\infty)\to[0,+\infty)$ and
$L^{\infty}_{\rm loc}$ function $c_{0}:\mathbb{R}\to[0,+\infty]$, such that

$\displaystyle
L(t,x,v)\geqslant\theta(|v|_{x})-c_{0},\qquad\forall(t,x,v)\in\mathbb{R}\times
TM.$

  3. (L3)

There exist $C_{1},C_{2}>0$, such that

$\displaystyle|L_{t}(t,x,v)|\leqslant
C_{1}+C_{2}L(t,x,v),\qquad\forall(t,x,v)\in\mathbb{R}\times TM.$

We call the Lagrangian $H$ associated with $L$ a _Tonelli Hamiltonian_ ,
defined by

$\displaystyle H(t,x,p)=\sup_{v\in
T_{x}M}\\{p(v)-L(t,x,v)\\},\qquad(t,x,p)\in\mathbb{R}\times T^{*}M.$

Given any $x,y\in M$, and $t_{1}<t_{2}$, we denote by
$\Gamma_{x,y}^{t_{1},t_{2}}$ the set of absolutely continuous curves
$\gamma\in\text{AC}([t_{1},t_{2}],M)$ with $\gamma(t_{1})=x$ and
$\gamma(t_{2})=y$.

######  Definition 2.4.

Given a Tonelli Hamiltonian $H$ with $L$ the associated Tonelli Lagrangian. We
call the function $h=h(t_{1},t_{2},x,y):\mathbb{R}\times\mathbb{R}\times
M\times M\to\mathbb{R}$ the _fundamental solution_ of the associated Hamilton-
Jacobi equation, defined by

$\displaystyle
h(t_{1},t_{2},x,y):=\inf_{\gamma\in\Gamma_{x,y}^{t_{1},t_{2}}}\int_{t_{1}}^{t_{2}}L(s,\gamma(s),\dot{\gamma}(s))\,ds,\qquad
t_{1}<t_{2},\quad x,y\in M.$

######  Definition 2.5.

Given a Tonelli Hamiltonian $H$ with $L$ the associated Tonelli Lagrangian.
For any $\phi:M\to\mathbb{R}$, $x\in M$, and $t_{1}<t_{2}$, we define

$\displaystyle T_{t_{1}}^{t_{2}}\phi(x):=\inf_{y\in
M}\\{\phi(y)+h(t_{1},t_{2},y,x)\\},$
$\displaystyle\breve{T}_{t_{1}}^{t_{2}}\phi(x):=\sup_{y\in
M}\\{\phi(y)-h(t_{1},t_{2},x,y)\\}.$

The families $\\{T_{t_{1}}^{t_{2}}\\}$ and $\\{\breve{T}_{t_{1}}^{t_{2}}\\}$
of operators are called _negative and positive Lax-Oleinik evolution_
respectively.

######  Remark 2.6.

If $\phi\in\text{\rm BUC}\,(M)$, the set of (bounded) uniformly continuous
functions on $M$, both $\\{T_{t_{1}}^{t_{2}}\\}$ and
$\\{\breve{T}_{t_{1}}^{t_{2}}\\}$ satisfy Markov property. Moreover, if $L$ is
independent of $t$, then both of them are semigroups on $\text{\rm BUC}\,(M)$.
We denote $T_{t}^{-}:=T_{0}^{t}$ and $T_{t}^{+}:=\breve{T}_{0}^{t}$ for
brevity.

##  3\. Optimal transport and singularities of $c$-concave function

Let $X$ and $Y$ be two Polish spaces and $c:X\times Y\to\mathbb{R}$. The
function $c$ is called _cost function_. The problem when
$c=h(t_{1},t_{2},\cdot,\cdot)$ for some fixed $t_{1}<t_{2}$ has very natural
feature. The basic relation among optimal transport, Hamilton-Jacobi
equations, Mather theory and weak KAM theory in this context has been
understood by many authors ([9, 10, 23]). Main part of this section is
motivated by our recent work on the Lax-Oleinik commutators and singularities.
We begin with the analysis without touching the measure theoretic aspect of
the theory at first.

###  3.1. Abstract Lax-Oleinik commutators

######  Definition 3.1.

For any $\phi:X\to\mathbb{R}$ and $\psi:Y\to\mathbb{R}$ we define the
_abstract Lax-Oleinik operators_

$\displaystyle T^{-}\phi(y):=\inf_{x\in X}\\{\phi(x)+c(x,y)\\},\qquad
T^{+}\psi(x):=\sup_{y\in Y}\\{\psi(y)-c(x,y)\\},\qquad x\in X,y\in Y.$

We always assume the infimum and supremum in the definitions of $T^{-}\phi(y)$
and $T^{+}\psi(x)$ can be achieved respectively.

######  Definition 3.2.

A function $\psi:Y\to[-\infty,+\infty)$ is said to be _$c$ -concave_ if it is
the infimum of a family of functions $c(x,\cdot)+\alpha(x)$. Analogously,
$\phi:X\to(-\infty,+\infty]$ is said to be _$c$ -convex_ if it is the supreme
of a family of functions $\beta(y)-c(\cdot,y)$.

It is clear that $T^{-}\phi$ is $c$-concave and $T^{+}\psi$ is $c$-convex.

######  Definition 3.3.

Given $\phi:X\to(-\infty,+\infty]$ and $\phi(x)<+\infty$ we define the
$c$-subdifferential as

$\displaystyle\partial_{c}\phi(x):=\\{y\in Y:\phi(\cdot)+c(\cdot,y)\
\text{attains minimum at}\ x\\}.$

Analogously, given $\psi:Y\to[-\infty,+\infty)$ and $\psi(y)>-\infty$ we
define the $c$-superdifferential as

$\displaystyle\partial^{c}\psi(y):=\\{x\in X:\psi(\cdot)-c(x,\cdot)\
\text{attains maximum at}\ y\\}.$

Now, we will concentrate on the Lax-Oleinik commutators $T^{-}\circ T^{+}$ and
$T^{+}\circ T^{-}$.

######  Lemma 3.4.

Let $\phi:X\to\mathbb{R}$ and $\psi:Y\to\mathbb{R}$.

  1. (1)

$T^{-}\circ T^{+}\psi\geqslant\psi$ and $T^{+}\circ T^{-}\phi\leqslant\phi$.

  2. (2)

$T^{-}\circ T^{+}\circ T^{-}\phi=T^{-}\phi$ and $T^{+}\circ T^{-}\circ
T^{+}\psi=T^{+}\psi$. In particular, both $T^{-}\circ T^{+}$ and $T^{-}\circ
T^{+}$ are idempotent.

###### Proof.

From the definition we have

$\displaystyle T^{-}\circ T^{+}\psi(y)=$ $\displaystyle\,\inf_{x\in
X}\\{\sup_{z\in Y}(\psi(z)-c(x,z))+c(x,y)\\}\geqslant\inf_{x\in
X}\\{\psi(y)-c(x,y)+c(x,y)\\}=\psi(y),$ $\displaystyle T^{+}\circ
T^{-}\phi(x)=$ $\displaystyle\,\sup_{y\in Y}\\{\inf_{z\in
X}(\phi(z)+c(z,y))-c(x,y)\\}\leqslant\sup_{y\in
Y}\\{\phi(x)+c(x,y)-c(x,y)\\}=\phi(x),$

and (1) follows. By (1) we observe

$\displaystyle T^{-}\phi\leqslant T^{-}\circ T^{+}\circ T^{-}\phi\leqslant
T^{-}\phi.$

Thus, the inequalities above are equalities and this completes the proof of
the first equality in (2). The proof of the rest of (2) is similar. ∎

######  Lemma 3.5.

  1. (1)

For any $y\in Y$, $T^{-}\circ T^{+}\psi(y)=\psi(y)$ if and only if
$\partial^{c}\psi(y)\not=\varnothing$.

  2. (2)

For any $x\in X$, $T^{+}\circ T^{-}\phi(x)=\phi(x)$ if and only if
$\partial_{c}\phi(x)\not=\varnothing$.

###### Proof.

We only prove (1) since the proof of (2) is similar. If
$\partial^{c}\psi(y)\not=\varnothing$, then there exists $x\in X$ such that
$T^{+}\psi(x)=\psi(y)-c(x,y)$. Thus,

$\displaystyle\psi(y)=T^{+}\psi(x)+c(x,y)\geqslant T^{-}\circ T^{+}\psi(y),$

and we have $T^{-}\circ T^{+}\psi(y)=\psi(y)$ since the converse inequality
holds automatically by Lemma 3.4. Now, suppose $T^{-}\circ
T^{+}\psi(y)=\psi(y)$, then there exists $x\in X$ such that
$\psi(y)=T^{-}\circ T^{+}\psi(y)=T^{+}\psi(x)+c(x,y)$. This implies
$\partial^{c}\psi(y)\not=\varnothing$. ∎

######  Theorem 3.6.

The following statements are equivalent.

  1. (1)

$T^{-}\circ T^{+}\psi=\psi$ (resp. $T^{+}\circ T^{-}\phi=\phi$).

  2. (2)

$\partial^{c}\psi(y)\not=\varnothing$ (resp.
$\partial_{c}\phi(x)\not=\varnothing$) for all $y\in Y$ (resp. $x\in X$).

  3. (3)

There exists $\phi:X\to\mathbb{R}$ (resp. $\psi:Y\to\mathbb{R}$) such that
$\psi=T^{-}\phi$ (resp. $\phi=T^{+}\psi$).

  4. (4)

$\psi$ (resp. $\phi$) is $c$-concave (resp. $c$-conconvex).

###### Proof.

The equivalence between (1) and (2) is immediately from Lemma 3.5. The
equivalence between (1) and (3) can be obtained by Lemma 3.4 (2).

If $T^{-}\circ T^{+}\psi=\psi$ then $\psi$ is $c$-concave by definition. Now,
suppose $\psi$ is $c$-concave which is the infimum of a family of $c$-affine
functions $c(x,\cdot)+\beta(x)$. That means $\psi=T^{-}\beta$. This leads to
$T^{-}\circ T^{+}\psi=\psi$ by the equivalence of (1) and (4). ∎

###  3.2. Singularities of $c$-concave functions

######  Definition 3.7.

For any $c$-concave function $\psi:Y\to\mathbb{R}$, $y\in Y$ is called a
_regular point_ of $\psi$ if $\partial^{c}\psi(y)$ is a singleton and is
called _singular point_ of $\psi$ if $\partial^{c}\psi(y)$ is not a singleton.
For any $c$-concave function $\psi:Y\to\mathbb{R}$, we denote by
$\text{Sing}^{c}(\psi)$ the set of all singular points of $\psi$.

######  Lemma 3.8.

Suppose $c$ is continuous. Then for any continuous $c$-concave function
$\psi:Y\to\mathbb{R}$, the set-valued map $y\mapsto\partial^{c}\psi(y)$ is
upper-semicontinuous.

###### Proof.

Suppose $\\{y_{k}\\}\subset Y$ is a sequence such that
$\lim_{k\to\infty}y_{k}=y$. Let $x_{k}\in\partial^{c}\psi(y_{k})$,
$k\in\mathbb{N}$, such that $x_{k}\to x$ as $k\to\infty$, i.e.,

$\displaystyle T^{+}\psi(x_{k})=\psi(y_{k})-c(x_{k},y_{k}),\qquad
k\in\mathbb{N}.$

Since $T^{+}\psi$ is lower-semicontinuous we have

$\displaystyle
T^{+}\psi(x)\geqslant\psi(y)-c(x,y)=\liminf_{k\to\infty}T^{+}\psi(x_{k})\geqslant
T^{+}\psi(x).$

Thus $T^{+}\psi(x)=\psi(y)-c(x,y)$ and this implies $x\in\partial^{c}\psi(y)$.
∎

######  Definition 3.9.

A point $x\in\partial^{c}\psi(y)$ is called a $c$-reachable gradient of $\psi$
at $y$ if there exists $y_{k}\to y$ as $k\to\infty$, each
$\partial^{c}\psi(y_{k})$ is a singleton, say
$\partial^{c}\psi(y_{k})=\\{x_{k}\\}$, and $\lim_{k\to\infty}x_{k}=x$. We
denote by $\partial^{*}_{c}\psi(y)\subset\partial^{c}\psi(y)$ the set of
$c$-reachable gradients of $\psi$ at $y$.

For the application of the singularities of $c$-concave functions to optimal
transport, we need clarify the difference of the singularities between usual
semiconcave function and $c$-concave function, even when $c$ is semiconcave.
For the rectifiability property of the singular set $\partial^{c}\psi$ of a
$c$-concave function under more general assumptions, the readers can refer to
[7]. We will discuss this problem for three types of specific cost functions
from weak KAM theory:

  1. 1.

$c(x,y)=h(t_{1},t_{2},x,y)$ with $t_{2}-t_{1}\ll 1$;

  2. 2.

$c(x,y)=h(t_{1},t_{2},x,y)$ with finite $t_{2}-t_{1}$;

  3. 3.

$c(x,y)=h(x,y)$ with $h$ the Peierls’ barrier.

Let $M$ be a connected and closed manifold and $\text{SCL}\,(M)$ be the set of
semiconcave functions on $M$. Now let us consider the case on the time-
independent Lagrangian $L$ and Hamiltonian $H$. For any
$\phi\in\text{SCL}\,(M)$, set

$\displaystyle t_{\phi}:=$ $\displaystyle\,\sup\\{t>0:T^{+}_{t}\phi\in
C^{1,1}(M)\\},$

where $T_{t}^{+}\phi:=\breve{T}_{0}^{t}\phi$. Moreover, we denote by

$\displaystyle\text{graph}\,(D^{+}\phi):=$ $\displaystyle\,\\{(x,p)\in
T^{*}M:x\in M,p\in D^{+}\phi(x)\\},$
$\displaystyle\text{graph}\,(DT_{t}^{+}\phi):=$
$\displaystyle\,\\{(x,DT_{t}^{+}\phi(x))\in T^{*}M:x\in M\\}.$

We call the set $\text{\rm graph}\,(D^{+}\phi)$ the _pseudo-graph_ of $D\phi$
or _1-graph_ of $\phi$. Now, we recall a result by Marie-Claude Arnaud ([6])
at first.

######  Proposition 3.10.

Let $\phi\in\text{\rm SCL}\,(M)$. Then $t_{\phi}>0$, and when $0<t<t_{\phi}$,
$T_{t}^{+}\phi\in C^{1,1}(M)$. Furthermore, for such $t\in(0,t_{\phi}]$,

$\text{\rm graph}\,(DT_{t}^{+}\phi)=\Phi_{H}^{-t}(\text{\rm
graph}\,(D^{+}\phi)),$ (3.1)

where $\\{\Phi^{s}_{H}\\}_{s\in\mathbb{R}}$ is the Hamiltonian flow with
respect to any Tonelli Hamiltonian $H:T^{*}M\to\mathbb{R}$.

######  Remark 3.11.

The original proof of Arnaud’s theorem is based on some approximation in the
sense of Hausdorff distance of the graph for time-independent Lagrangian (see
also [26] for the case of time dependent Lagrangian). The relation (3.1)
defines a diffeomorphism and bi-Lipschitz homeomorphism between $\text{\rm
graph}\,(D^{+}\phi)$ and $\text{\rm graph}\,(DT^{+}_{t}\phi)$ by the
Hamiltonian flow. This allows us to understand the singularities of $\phi$ by
means of the latter Lipschitz graph. Except for some finer results obtain in
[1], this idea also leads to more topological results of the singular set of
semiconcave functions and $c$-concave functions from the theory of optimal
transport ([26]).

Since Proposition 3.10 only holds for short time $t_{2}-t_{1}\ll 1$, we have
to consider our problem for $c(x,y)=h(t_{1},t_{2},x,y)$ in the following three
cases: $t_{2}-t_{1}\ll 1$, $t_{2}-t_{1}$ is finite and $t_{2}-t_{1}\to\infty$.

####  3.2.1. $c(x,y)=h(t_{1},t_{2},x,y)$ with $t_{2}-t_{1}\ll 1$

######  Proposition 3.12.

Let $M$ be a connected closed manifold and $\psi:M\to\mathbb{R}$ be a
$c$-concave function. Then $\psi$ be semiconcave in usual sense. For the cost
function $c(x,y)=h(t_{1},t_{2},x,y)$, $x,y\in M$, with
$0<t_{2}-t_{1}<t_{\psi}$ as in Proposition 3.10, there exists a bi-Lipschitz
homeomorphism between $\partial^{c}\psi(y)$ and $D^{+}\psi(y)$ for all $y\in
M$. In particular,

$\displaystyle\mbox{\rm Sing}^{c}(\psi)=\mbox{\rm Sing}\,(\psi),$

where $\mbox{\rm Sing}\,(\psi)$ is the set of the points of non-
differentiability of $\psi$.

###### Proof.

By Proposition 3.10 (see also [26, Theorem 3.14]), we have

$\displaystyle\Phi_{H}^{t_{1},t_{2}}(\text{\rm
graph}\,(D\breve{T}_{t_{1}}^{t_{2}}\psi))=\text{\rm graph}\,(D^{+}\psi),$

where $\\{\Phi^{t_{1},t_{2}}_{H}\\}$ is the non-autonomous Hamiltonian flow
from $t_{1}$ to $t_{2}$. Given $y\in M$, and $\forall p\in D^{+}\psi(y)$ we
denote by $(\gamma(\cdot),p(\cdot))$ the Hamiltonian trajectory determined by
endpoint condition $(\gamma(t_{2}),p(t_{2}))=(y,p)$. Recalling that
$p(t_{2})=L_{v}(t_{2},\gamma(t_{2}),\dot{\gamma}(t_{2}))\in D^{+}\psi(y)$, it
follows $0\in D^{+}(\psi(y)-h(t_{1},t_{2},\gamma(t_{1}),y))$ by Fermat rule,
or equivalently that $y$ is a maximizer of the function
$\phi(\cdot)-h(t_{1},t_{2},\gamma(t_{1}),\cdot)$. Therefore, for
$x=\gamma(t_{1})$ we have
$\breve{T}^{t_{2}}_{t_{1}}\psi(x)=\psi(y)-h(t_{1},t_{2},x,y)$ which implies
$x\in\partial^{c}\psi(y)$ since
$D\breve{T}^{t_{2}}_{t_{1}}\psi(x)=p(t_{1})=L_{v}(t_{1},\gamma(t_{1}),\dot{\gamma}(t_{1}))$.
The converse direction can be handled in a similar way. For any $y\in M$, set

$\displaystyle\Gamma(y):=\\{(x,p):x\in\partial^{c}\psi(y),p=D\breve{T}^{t_{2}}_{t_{1}}\psi(x)\\}\subset\text{\rm
graph}\,(D\breve{T}_{t_{1}}^{t_{2}}\psi).$

Then, this leads to the relation

$\displaystyle(y,D^{+}\psi(y))=\Phi_{H}^{t_{1},t_{2}}(\Gamma(y)),\qquad y\in
M.$

Notice that $\text{\rm graph}\,(D\breve{T}_{t_{1}}^{t_{2}}\psi)$ is the graph
of a Lipschitz function. This shows there is a bi-Lipschitz homeomorphism
between $\partial^{c}\psi(y)$ and $D^{+}\psi(y)$ for each $y\in M$. ∎

####  3.2.2. $c(x,y)=h(t_{1},t_{2},x,y)$ with finite $t_{2}-t_{1}$

######  Lemma 3.13.

Let $S$ be a compact topological space and
$F:S\times\mathbb{R}^{d}\to\mathbb{R}$. Suppose $F$ is continuous and
$F(s,\cdot)$ is uniformly semiconcave with constant $C\geqslant 0$ for $s\in
S$. Set $u:\mathbb{R}^{d}\to\mathbb{R}$,

$\displaystyle u(x)=\inf\\{F(s,x):s\in S\\},\qquad x\in\mathbb{R}^{d}.$

For any $x\in\mathbb{R}^{d}$, let $M(x)=\\{s\in S:F(s,x)=u(x)\\}$. Then $u$ is
semiconcave with constant $C$ and $D^{+}u(x)=\mbox{\rm
co}\,\\{D^{+}_{x}F(s,x):s\in M(x)\\}$.

Lemma 3.13 is a refinement of Proposition 2.2. We give a proof in the
appendix.

######  Proposition 3.14.

Under the same assumption of Proposition 3.12 except for that
$0<t_{2}-t_{1}<t_{\psi}$. Then the following statements hold.

  1. (1)

$D^{+}\psi(y)$ is a singleton implies $\partial^{c}\psi(y)$ is a singleton.
Consequently,

$\displaystyle\mathrm{Sing}^{c}(\psi)\subset\mathrm{Sing}\,(\psi).$

  2. (2)

There exists a Lipschitz map from $D^{*}\psi(y)$ to $\partial^{*}_{c}\psi(y)$.

###### Proof.

Let $y\in M$. We suppose $y$ is a point of differentiability of $\psi$. Since
$T_{t_{1}}^{t_{2}}\circ\breve{T}_{t_{1}}^{t_{2}}\psi=\psi$, then
$h(t_{1},t_{2},x,\cdot)$ is differentiable at $y$ and
$D_{y}h(t_{1},t_{2},x,y)$ coincides for any
$x\in\arg\min\\{\breve{T}_{t_{1}}^{t_{2}}\psi(\cdot)+h(t_{1},t_{2},\cdot,y)\\}$
by Lemma 3.13. Recall that each
$D_{y}h(t_{1},t_{2},x,y)=L_{v}(t_{2},\gamma_{x}(t_{2}),\dot{\gamma}_{x}(t_{2}))$
with $\gamma_{x}\in\Gamma^{t_{1},t_{2}}_{x,y}$ the unique minimizer for
$h(t_{1},t_{2},x,y)$. Since each $\gamma_{x}$ satisfies the same endpoint
condition

$\displaystyle\gamma_{x}(t_{2})=y,\qquad\dot{\gamma}_{x}(t_{2})=H_{p}(t_{2},y,D_{y}h(t_{1},t_{2},x,y)),$

we conclude that there exists a unique $x\in M$ such that

$\displaystyle\psi(y)=T_{t_{1}}^{t_{2}}\circ\breve{T}_{t_{1}}^{t_{2}}\psi(y)=\breve{T}_{t_{1}}^{t_{2}}\psi(x)+h(t_{1},t_{2},x,y)$

and $D\psi(y)=D_{y}h(t_{1},t_{2},x,y)$. The first part of (1) implies
$\text{Sing}^{c}(\psi)\subset\text{Sing}\,(\psi)$. This completes the proof of
(1).

Now, suppose $p\in D^{*}\psi(y)$. Then there exists a sequence $y_{k}\to y$ as
$k\to\infty$, $\psi$ is differentiable at each $y_{k}$ and
$p=\lim_{k\to\infty}D\psi(y_{k})$. From the first part of the proof, for each
$k$ there exists a unique $x_{k}$ such that
$\partial^{c}\psi(y_{k})=\\{x_{k}\\}$. As in the proof of the first part, let
$\gamma_{x_{k}}\in\Gamma^{t_{1},t_{2}}_{x_{k},y_{k}}$ be the unique minimal
curve for $h(t_{1},t_{2},x_{k},y_{k})$. Applying Ascoli-Arzela theorem, any
convergent subsequence of $\\{\gamma_{x_{k}}\\}$ has a limiting curve
$\gamma\in\Gamma^{t_{1},t_{2}}_{x,y}$ under $C^{2}$-topology satisfying the
same endpoint condition $\gamma(t_{2})=y$ and
$L_{v}(t_{2},\gamma(t_{2}),\dot{\gamma}(t_{2}))=p$. This implies the sequence
$\\{\gamma_{x_{k}}\\}$ converges a curve $\gamma:[t_{1},t_{2}]\to M$
connecting $x$ to $y$. By continuity, we have

$\displaystyle\psi(y)=\breve{T}_{t_{1}}^{t_{2}}\psi(\gamma(0))+\int^{t_{2}}_{t_{1}}L(s,\gamma(s),\dot{\gamma}(s))\
ds.$

This implies $\lim_{k\to\infty}x_{k}=x$ and $x\in\partial^{*}_{c}\psi(y)$. Let
$i$ be the inclusion from $D^{*}\psi(y)$ to the graph $\\{(y,p):p\in
D^{*}\psi(y)\\}$. We finish the proof by observing that
$\pi_{x}\circ\Phi^{t_{1},t_{2}}_{H}\circ i$ defines a Lipschitz map from
$D^{*}\psi(y)$ to $\partial^{*}_{c}\psi(y)$. ∎

####  3.2.3. $c(x,y)=h(x,y)$ with $h$ the Peierls’ barrier

Now, consider the case when $L$ is time-independent and the cost function
$c(x,y)=h(x,y)$, where $h(x,y)=\liminf_{t\to\infty}h(0,t,x,y)+c[0]t$ is the
Peierls’ barrier ([22]). In [25], Mather introduced a pseudo-distance on the
Aubry set. We suppose $c[0]=0$ and let

$\displaystyle d(x,y)=h(x,y)+h(y,x),\qquad\forall x,y\in\mathcal{A}.$

Mather proved $d$ is a pseudo-distance on $\mathcal{A}$ and each equivalence
class of the relation $d(x,y)=0$ is called a static class (see more details in
[21]).

######  Proposition 3.15.

Let $L$ be a time-independent Tonelli Lagrangian, $u^{-}$ be a weak KAM
solution and and $c=h$ with $h$ the Peierls’ barrier. Then, for any $y\in M$,
there exists a static class $\mathcal{A}_{y}\subset\partial^{c}u^{-}(y)$.

###### Proof.

Recall that $u^{-}=T^{-}_{t}\circ T^{+}_{t}u^{-}$ for all $t\geqslant 0$
([17]), where $T_{t}^{-}\phi:=T_{0}^{t}\phi$. Fix $y\in M$ and let $p\in
D^{*}u^{-}(y)=D^{*}T^{-}_{t}\circ T^{+}_{t}u^{-}(y)$. Then there exists
$\gamma_{t}:[-t,0]\to M$ with $\gamma_{t}(0)=y$ and
$L_{v}(\gamma_{t}(0),\dot{\gamma}_{t}(0))=p$ such that

$\displaystyle u^{-}(y)=T^{-}_{t}\circ
T^{+}_{t}u^{-}(y)=T^{+}_{t}u^{-}(\gamma_{t}(-t))+\int^{0}_{-t}L(\gamma_{t}(s),\dot{\gamma}_{t}(s))\
ds=T^{+}_{t}u^{-}(\gamma_{t}(-t))+h(-t,0,\gamma_{t}(-t),y).$

Observe that since all the curves $\gamma_{t}$ are extremals with the same
endpoint condition. Thus, there exists $\gamma:(-\infty,0]\to M$ with
$\gamma(0)=y$ and $L_{v}(\gamma(0),\dot{\gamma}(0))=p$ such that
$\gamma_{t}=\gamma|_{[-t,0]}$. From classical weak KAM theory we know that
$T^{+}_{t}u^{-}$ converges uniformly to $u^{+}$, the conjugate function of
$u^{-}$, as $t\to+\infty$. Then, for any limiting point $x$ of $\gamma(-t)$ as
$t\to+\infty$, we have

$\displaystyle u^{-}(y)=u^{+}(x)+h(x,y).$

Such points $x\in\mathcal{A}$, the projected Aubry set.

For any $x,x^{\prime}\in\mathcal{A}$ belonging in the same static class we
have

$\displaystyle h(x,x^{\prime})+h(x^{\prime},x)=0,$

and

$\displaystyle u^{+}(x^{\prime})-u^{+}(x)=h(x,x^{\prime}).$

Therefore, if $x\in\mathcal{A}$ is given in the first part of the proof, which
satisfies $u^{+}(x)=u^{-}(y)-h(x,y)$, then for any $x^{\prime}\in\mathcal{A}$
belonging in the same static class as $x$ we have

$\displaystyle
u^{+}(x^{\prime})=u^{+}(x)+h(x,x^{\prime})=u^{-}(y)-h(x,y)-h(x^{\prime},x).$

By the property of Peierls’ barrier

$\displaystyle h(x^{\prime},y)\leqslant h(x^{\prime},x)+h(x,y).$

It follows

$\displaystyle u^{+}(x^{\prime})\leqslant
u^{-}(y)-h(x^{\prime},y),\qquad\text{or}\qquad
u^{-}(y)-u^{+}(x^{\prime})\geqslant h(x^{\prime},y).$

The second inequality is indeed an equality because $(u^{-},u^{+})$ is an
admissible Kantorovich pair. This completes the proof. ∎

The following corollary is immediate by Proposition 3.15 and the definition of
$\text{Sing}^{c}(u^{-})$.

######  Corollary 3.16.

If each static class is not a singleton, then $\mbox{\rm Sing}^{c}(u^{-})=M$.

##  4\. Application of abstract Lax-Oleinik operators to optimal transport

###  4.1. An alternative formulation of Kantorovich-Rubinstein duality

For any Polish space $X$ let $\mathscr{P}(X)$ be the space of Borel
probability measures on $X$. Given two Polish spaces $X$ and $Y$, we call
$\pi\in\mathscr{P}(X\times Y)$ a transport plan between $\mu\in\mathscr{P}(X)$
and $\nu\in\mathscr{P}(X)$ if

$\displaystyle(p_{X})_{\\#}\pi=\mu,\qquad(p_{Y})_{\\#}\pi=\nu$

where $p_{X}$ and $p_{Y}$ are the projection from $X\times Y$ to $X$ and $Y$
respectively. We denote by $\Gamma(\mu,\nu)$ the set of all transport plans
between $\mu$ and $\nu$.

For any Borel cost function $c:X\times Y\to\overline{\mathbb{R}}$ uniformly
bounded below, we consider the following problem of Kantorovich

$\mathcal{C}(\mu,\nu):=\inf\Big{\\{}\int_{X\times Y}c(x,y)\
d\pi:\pi\in\Gamma(\mu,\nu)\Big{\\}}$ (K)

We denote by $\Gamma_{o}(\mu,\nu)$ the set of optimal transport plans.

To formulate the important theorem of Kantorovich-Rubinstein duality, we
introduce

$\displaystyle I_{c}:=$
$\displaystyle\,\\{(\phi,\psi):\psi(y)-\phi(x)\leqslant c(x,y)\ \text{for
all}\ x\in X,y\in Y\\},$ $\displaystyle K_{c}:=$
$\displaystyle\,\\{(\phi,\psi)\in I_{c}:\psi=T^{-}\phi,\phi=T^{+}\psi\\}.$

It is obvious that

$\displaystyle I_{c}=\\{(\phi,\psi):\psi\leqslant T^{-}\phi,\phi\geqslant
T^{+}\psi\\}\supset K_{c}.$

In the literature, the elements in $K_{c}$ is called _admissible Kantorovich
pair_.

######  Theorem 4.1 (Kantorovich-Rubinstein duality).

$\mathcal{C}(\mu,\nu)=\sup_{(\phi,\psi)\in I_{c}}\Big{\\{}\int_{Y}\psi\
d\nu-\int_{X}\phi\ d\mu\Big{\\}}=\sup_{(\phi,\psi)\in
K_{c}}\Big{\\{}\int_{Y}\psi\ d\nu-\int_{X}\phi\ d\mu\Big{\\}}.$ (KR)

For our purpose, we try to reformulate Theorem 4.1 and connect it directly to
the Lax-Oleinik operators $T^{\pm}$. Let $\phi:X\to\mathbb{R}$ and
$\psi:Y\to\mathbb{R}$. Set $c_{\phi}(x,y):=\phi(x)+c(x,y)$ and
$c^{\psi}(x,y)=\psi(y)-c(x,y)$. For any $\mu\in\mathscr{P}(X)$ and
$\nu\in\mathscr{P}(Y)$, we try to find a function $\phi:X\to\mathbb{R}$ and a
function $\psi:Y\to\mathbb{R}$ such that

$\displaystyle\int_{Y}T^{-}\phi\ d\nu=$
$\displaystyle\,\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times Y}c_{\phi}(x,y)\
d\pi,$ (K-) $\displaystyle\int_{Y}T^{+}\psi\ d\mu=$
$\displaystyle\,\sup_{\pi\in\Gamma(\mu,\nu)}\int_{X\times Y}c^{\psi}(x,y)\
d\pi.$ (K+)

######  Theorem 4.2.

  1. (1)

If $\phi:X\to\mathbb{R}$ is a solution of (K-), then $(\phi,T^{-}\phi)\in
I_{c}$ is a solution of (KR). Conversely, if $(\phi,\psi)\in I_{c}$ is a
solution of (KR), then $\phi$ solves (K-).

  2. (2)

If $\psi:Y\to\mathbb{R}$ is a solution of (K+), then $(T^{+}\psi,\psi)\in
I_{c}$ is a solution of (KR). Conversely, if $(\phi,\psi)\in I_{c}$ is a
solution of (KR), then $\psi$ solves (K+).

###### Proof.

For any $\phi:X\to\mathbb{R}$ we observe that

$\int_{Y}T^{-}\phi\ d\nu\leqslant\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times
Y}c_{\phi}(x,y)\ d\pi,$ (4.1)

since for any $\pi\in\Gamma(\mu,\nu)$,

$\displaystyle\int_{Y}T^{-}\phi\ d\nu=\int_{Y}T^{-}\phi\
d\pi\leqslant\int_{X\times Y}\phi(x)+c(x,y)\ d\pi(x,y)=\int_{X}\phi\
d\mu+\int_{X\times Y}c(x,y)\ d\pi.$

Thus inequality (4.1) reads

$\displaystyle\int_{Y}T^{-}\phi\ d\nu-\int_{X}\phi\
d\mu\leqslant\mathcal{C}(\mu,\nu).$

Since $\phi$ is a solution of (K-), then the inequality above is indeed an
equality. Notice $(\phi,T^{-}\phi)\in I_{c}$ since
$T^{-}\phi(y)-\phi(x)\leqslant c(x,y)$. This implies $(\phi,T^{-}\phi)$ is a
solution of (KR) by Theorem 4.1.

Conversely, suppose $(\phi,\psi)\in I_{c}$ is a solution of (KR). Then

$\displaystyle\int_{Y}\psi\ d\nu\geqslant\int_{X}\phi\ d\mu+\int_{X\times
Y}c(x,y)\ d\pi=\int_{X\times Y}c_{\phi}(x,y)\ d\pi.$

Since $\psi\leqslant T^{-}\phi$, then $(\phi,T^{-}\phi)$ is also a solution of
(KR). Together with (4.1), $\phi$ is a solution of (K-). This completes the
proof of (1). The proof (2) is similar. ∎

###  4.2. Mass transport aspects of Lax-Oleinik operator

######  Lemma 4.3.

Assume that $\pi\in\mathscr{P}(X\times Y)$, then

  1. (1)

$p_{x}(\mathrm{supp}\,(\pi))\subset\mathrm{supp}\,\,((p_{x})_{\\#}\pi)$;

  2. (2)

$p_{x}(\mathrm{supp}\,(\pi))=\mathrm{supp}\,((p_{x})_{\\#}\pi)$ if $Y$ is
compact.

######  Remark 4.4.

The compactness of $Y$ in Lemma 4.3 (2) is essential. Assume that
$X=Y=\mathbb{R}$ and
$\pi=\Sigma_{i=1}^{+\infty}2^{-i}\delta_{(\frac{1}{i},i)}\in\mathscr{P}(\mathbb{R}^{2})$,
then

$\displaystyle\mathrm{supp}\,(\pi)=\bigg{\\{}\big{(}\frac{1}{i},i\big{)}:i\in\mathbb{N}\bigg{\\}},\qquad
p_{x}(\mathrm{supp}\,\,(\pi))=\bigg{\\{}\frac{1}{i}:i\in\mathbb{N}\bigg{\\}}.$

However,

$\displaystyle(p_{x})_{\\#}\pi=\Sigma_{i=1}^{+\infty}2^{-i}\delta_{\frac{1}{i}},\qquad\mathrm{supp}\,\,((p_{x})_{\\#}\pi)=\bigg{\\{}\frac{1}{i}:i\in\mathbb{N}\bigg{\\}}\cup\\{0\\}.$

######  Lemma 4.5.

Suppose $\mu\in\mathscr{P}(X)$ and $\nu\in\mathscr{P}(Y)$. Then
$\phi\in\emph{Lip}_{b}(X)$ is a solution of (K-) if and only if for all
$\pi\in\Gamma_{o}(\mu,\nu)$,

$\mathrm{supp}\,(\pi)\subset\\{(x,y)\in X\times
Y:x=\arg\min\\{\phi(\cdot)+c(\cdot,y)\\}\\}=\\{(x,y):y\in\partial_{c}\phi(x)\\}.$
(4.2)

###### Proof.

For $\phi\in\text{\rm Lip}_{b}(X)$ let

$\displaystyle F_{\phi}(x,y):=\phi(x)+c(x,y)-T^{-}\phi(y),\qquad(x,y)\in
X\times Y$

By the definition of $T^{-}\phi$, $F_{\phi}(x,y)\geqslant 0$ for all $(x,y)\in
X\times Y$. $T^{-}\phi$ is upper semi-continuous, and then $F_{\phi}(x,y)$ is
a lower semi-continuous.

If (K-) holds true, then for any optimal plan $\pi\in\Gamma_{o}(\mu,\nu)$,

$\int_{X\times Y}F_{\phi}(x,y)\,d\pi=0.$ (4.3)

Let $\pi\in\Gamma_{o}(\mu,\nu)$ and
$(x_{0},y_{0})\in\mathrm{supp}\,(\pi)\setminus\\{(x,y)|y\in\partial_{c}\phi(x)\\}$.
Thus $F_{\phi}(x_{0},y_{0})>0$. Due to the lower semi-continuity of
$F_{\phi}$, for arbitrary $\varepsilon\in(0,F_{\phi}(x_{0},y_{0}))$, there
exists $\delta>0$ such that if $(x,y)\in B((x_{0},y_{0}),\delta)$ then

$\displaystyle F_{\phi}(x,y)>F_{\phi}(x_{0},y_{0})-\varepsilon>0.$

Since $\pi(B((x_{0},y_{0}),\delta))>0$, it follows

$\displaystyle\int_{X\times
Y}F_{\phi}(x,y)\,d\pi\geqslant\int_{B((x_{0},y_{0}),\delta)}F_{\phi}(x,y)\,d\pi>\int_{B((x_{0},y_{0}),\delta)}F_{\phi}(x_{0},y_{0})-\varepsilon\,d\pi>0,$

which contradicts to (4.3).

Conversely, suppose (4.2) holds true for all $\pi\in\Gamma_{o}(\mu,\nu)$.
According to the disintegration of the measure $\pi=\int_{Y}\pi_{y}\,d\nu$, we
obtain

$\displaystyle\int_{X\times Y}\phi(x)+c(x,y)\,d\pi$
$\displaystyle=\int_{X\times
Y}[\phi(x)+c(x,y)]\cdot\mathbf{1}_{\\{(x,y):y\in\partial_{c}\phi(x)\\}}(x,y)\,d\pi$
$\displaystyle=\int_{Y}\int_{\\{(x,y):y\in\partial_{c}\phi(x)\\}}[\phi(x)+c(x,y)]\,d\pi_{y}d\nu.$

Recall that $T^{-}\phi(y)\equiv\phi(x)+c(x,y)$ for all $(x,y)$ such that
$y\in\partial_{c}\phi(x)$. Thus,

$\displaystyle\int_{X\times
Y}\phi(x)+c(x,y)\,d\pi=\int_{Y}\int_{\\{(x,y):y\in\partial_{c}\phi(x)\\}}\phi(x)+c(x,y)\,d\pi_{y}d\nu=\int_{Y}T^{-}\phi(y)\,d\nu,$

which means $\phi$ is a solution of (K-). ∎

From now, we assume that $X=Y=M$, a connected and closed manifold of dimension
$d$. For technical reason we suppose that $TM$ and $T^{*}M$, the associated
tangent space and cotangent space of $M$ respectively, has a trivialization.
We consider the cost function as $c(x,y)=h(0,t,x,y)$ for some fixed $t>0$.
Recall that $T^{-}=T_{0}^{t}$ and $T^{+}=\breve{T}_{0}^{t}$.

######  Proposition 4.6.

Let $y_{0}\in M$, $\phi\in\emph{Lip}(M)$ and $\nu=\delta_{y_{0}}$, then (4.2)
holds if and only if

$\mathrm{supp}\,\,(\mu)\subset\\{x:y_{0}\in\partial_{c}\phi(x)\\}.$ (4.4)

In particular, if $\phi$ is $c$-convex, then
$\\{x:y_{0}\in\partial_{c}\phi(x)\\}=\partial^{c}T^{-}\phi(y_{0})$.

###### Proof.

Given $\in\text{Lip}(M)$ and $\pi\in\Gamma_{o}(\mu,\nu)$ such that (4.2)
holds. Since $\nu=\delta_{y_{0}}$, by Lemma 4.3 we have

$\displaystyle
p_{y}(\mathrm{supp}\,\,(\pi))\subset\mathrm{supp}\,\,((p_{y})_{\\#}\pi)=\mathrm{supp}\,\,(\delta_{y_{0}})=\\{y_{0}\\}.$

Together with Lemma 4.5,
$\mathrm{supp}\,\,(\pi)\subset\\{(x,y_{0}):y_{0}\in\partial_{c}\phi(x)\\}$.
Then, owing to the compactness of $M$, Lemma 4.3 (2) implies that

$\displaystyle\mathrm{supp}\,(\mu)=\mathrm{supp}\,\,((p_{x})_{\\#}\pi)\subset
p_{x}(\\{(x,y_{0}):y_{0}\in\partial_{c}\phi(x)\\})=\\{x:y_{0}\in\partial_{c}\phi(x)\\}.$

Conversely, if (4.4) holds, according to Lemma 4.3,

$\displaystyle
p_{x}(\mathrm{supp}\,\,(\pi))\subset\mathrm{supp}\,((p_{x})_{\\#}\pi)=\mathrm{supp}\,\,(\mu)\subset\\{x:y_{0}\in\partial_{c}\phi(x)\\},$

which implies that

$\displaystyle\mathrm{supp}\,(\pi)\subset
p_{x}^{-1}(\\{x:y_{0}\in\partial_{c}\phi(x)\\})=\\{x:y_{0}\in\partial_{c}\phi(x)\\}\times
M.$

On the other hand,

$\displaystyle
p_{y}(\mathrm{supp}\,\,(\pi))\subset\mathrm{supp}\,((p_{y})_{\\#}\pi)=\mathrm{supp}\,(\delta_{y_{0}})=\\{y_{0}\\}.$

Thus, from the analysis above,

$\displaystyle\mathrm{supp}\,(\pi)\subset\\{(x,y_{0}):y_{0}\in\partial_{c}\phi(x)\\}\subset\\{(x,y):y\in\partial_{c}\phi(x)\\}$

and this completes the proof together with Lemma 4.5. ∎

######  Proposition 4.7.

Assume that $y_{0}\in M$, $\nu=\delta_{y_{0}}$, $\phi\in\emph{Lip}(M)$ and
$\\{\Phi_{H}^{t_{1},t_{2}}\\}_{t_{1},t_{2}\in\mathbb{R}}$ is the associated
Hamiltonian flow from $t_{1}$ to $t_{2}$.

  1. (1)

Then for any $\rho\in\mathscr{P}(\mathbb{R}^{d})$ with
$\mathrm{supp}\,(\rho)\subset D^{*}T^{-}\phi(y_{0})$, there exists
$\mu_{\rho}:=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho)$,
which makes (K-) holds true;

  2. (2)

For each $\mu\in\mathscr{P}(M)$ which satisfies (K-), there exists
$\rho_{\mu}\in\mathscr{P}(\mathbb{R}^{d})$ with
$\mathrm{supp}\,(\rho_{\mu})\subset D^{+}T^{-}\phi(y_{0})$, such that
$\mu=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho_{\mu})$.

###### Proof.

Note that there exists $K>0$, such that
$\Phi_{H}^{t,0}(M\times[-\text{Lip}\,(\phi),\text{Lip}(\phi)]^{d})\subset
M\times[-K,K]^{d}$. Since $[-K,K]^{d}$ is compact, by Lemma 4.3, we only need
to check

$\displaystyle\mathrm{supp}\,(\mu_{\rho})=\tilde{p}_{x}(\mathrm{supp}\,((\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho)))\subset\\{x:y_{0}\in\partial_{c}\phi(x)\\},$

where $\tilde{p}_{x}:M\times[-K,K]^{d}\subset T^{*}M\to M$,
$\tilde{p}_{x}(x,p)=x$.

Suppose
$x_{0}\in\mathrm{supp}\,((\tilde{p}_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho))$.
Then, there exists some $p_{0}\in[-K,K]^{d}$ such that
$(x_{0},p_{0})\in\mathrm{supp}\,\,((\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho))$,
which means for any $r>0$ and $B((x_{0},p_{0}),r)\subset T^{*}M$,

$0<[(\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho)]\,(B((x_{0},p_{0}),r))=(\delta_{y_{0}}\times\rho)\,(\Phi_{H}^{0,t}(B(x_{0},p_{0}),r)).$
(4.5)

We claim that $\Phi_{H}^{0,t}(x_{0},p_{0})\in\\{y_{0}\\}\times
D^{*}T^{-}\phi(y_{0})$. Otherwise, by the fact
$\mathrm{supp}\,\,(\delta_{y_{0}}\times\rho)\subset\\{y_{0}\\}\times
D^{*}T^{-}\phi(y_{0})$,
$\Phi_{H}^{0,t}(x_{0},p_{0})\notin\mathrm{supp}\,\,(\delta_{y_{0}}\times\rho)$.
This contradicts to (4.5). Thus, there exists a unique $p^{*}\in
D^{*}T^{-}\phi(y_{0})$, $\Phi_{H}^{0,t}(x_{0},p_{0})=(y_{0},p^{*})$. By
Theorem 6.4.9 in [19], if we let
$\tilde{p}_{x}\circ\Phi_{H}^{0,s}(x_{0},p_{0})=\xi(s)$, $s\in[0,t]$, then

$\displaystyle T^{-}\phi(y_{0})$
$\displaystyle=\inf_{\gamma(t)=y_{0}}\left\\{\phi(\gamma(0))+\int_{0}^{t}L(s,\gamma(s),\dot{\gamma}(s))\,ds\right\\}$
$\displaystyle=\phi(x_{0})+\int_{0}^{t}L(s,\xi(s),\dot{\xi}(s))\,ds$
$\displaystyle=\phi(x_{0})+c(x_{0},y_{0}),$

which means $y_{0}\in\partial_{c}\phi(x_{0})$. In summary,
$\mu_{\rho}=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{0}}\times\rho)$
satisfies (4.4) and (K-) holds by Lemma 4.5 and Proposition 4.6. This
completes the proof of (1).

For the second part of the proof, we consider the following set-valued map

$\displaystyle\Gamma:\\{x:y_{0}\in\partial_{c}\phi(x)\\}$ $\displaystyle\to
T^{*}M$ $\displaystyle x$ $\displaystyle\rightsquigarrow(x,P(x)),$

where $P(x):=\\{p:(x,p)\in\Phi_{H}^{t,0}(y_{0},D^{+}T^{-}\phi(y_{0}))\\}$. Due
to the closedness of $D^{+}T^{-}\phi(y_{0}))$, it is easy to check that $P(x)$
is a non-empty closed set for any $x$. Moreover, for each open subset
$U\subset\mathbb{R}^{d}$,

$\displaystyle\\{x:P(x)\cap U\neq\varnothing\\}$ $\displaystyle=\\{x:\exists
p\in U,(x,p)\in\Phi_{H}^{t,0}(y_{0},D^{+}T^{-}\phi(y_{0}))\\}$
$\displaystyle=p_{x}(\Phi_{H}^{t,0}(y_{0},D^{+}T^{-}\phi(y_{0}))\cap(U\times\\{x:y_{0}\in\partial_{c}\phi(x)\\}))$

is a measurable subset of $M$. By a standard measurable selection theorem
(see, for example, [20]), there exists a measurable selection
$\tilde{\Gamma}(x):=(x,\tilde{P}(x))\in(x,P(x))$. Set
$\tilde{\nu}:=(\Phi_{H}^{0,t}\circ\tilde{\Gamma})_{\\#}\mu$.

We claim that
$\mathrm{supp}\,(\tilde{\nu})\subset(y_{0},D^{+}T^{-}\phi(y_{0}))$. If not, we
assume that
$(y_{*},p_{*})\in\mathrm{supp}\,(\tilde{\nu})\setminus(y_{0},D^{+}T^{-}\phi(y_{0}))$.
Then there exists $r_{0}>0$ such that
$B((y_{*},p_{*}),r_{0})\cap(y_{0},D^{+}T^{-}\phi(y_{0}))=\varnothing$, which
means

$\displaystyle\Phi_{H}^{t,0}(B((y_{*},p_{*}),r_{0})\cap\Phi_{H}^{t,0}((y_{0},D^{+}T^{-}\phi(y_{0}))=\varnothing.$

Since $\Phi_{H}^{t,0}$ is a diffeomorphism, according to the definition of
$\tilde{\Gamma}$,
$\tilde{\Gamma}^{-1}(\Phi_{H}^{t,0}(B((y_{*},p_{*}),r_{0}))=\varnothing$.
Moreover, $\tilde{\nu}(B((y_{*},p_{*}),r_{0}))>0$, because
$(y_{*},p_{*})\in\mathrm{supp}\,(\tilde{\nu})$. However,

$\displaystyle
0<\tilde{\nu}(B((y_{*},p_{*}),r_{0}))=(\Phi_{H}^{0,t}\circ\tilde{\Gamma})_{\\#}\mu(B((y_{*},p_{*}),r_{0}))=\mu(\tilde{\Gamma}^{-1}(\Phi_{H}^{t,0}(B((y_{*},p_{*}),r_{0}))))=\mu(\varnothing)=0.$

This leads to a contradiction and it means
$\mathrm{supp}\,(\tilde{\nu})\subset(y_{0},D^{+}T^{-}\phi(y_{0}))$.

In addition, $(p_{x})_{\\#}\tilde{\nu}=\delta_{\\{y_{0}\\}}$. Invoking the
disintegration theorem, we assert that there exists
$\tilde{\nu}_{y_{0}}\in\mathscr{P}(\mathbb{R}^{d})$ with
$\tilde{\nu}=\delta_{\\{y_{0}\\}}\times\tilde{\nu}_{y_{0}}$. Thus,
$\mathrm{supp}\,(\tilde{\nu}_{y_{0}})\subset D^{+}T^{-}\phi(y_{0})$ since
$\mathrm{supp}\,(\tilde{\nu})\subset(y_{0},D^{+}T^{-}\phi(y_{0}))$. We
complete the proof by choosing $\rho_{\mu}:=\tilde{\nu}_{y_{0}}$. ∎

######  Lemma 4.8.

Let $\\{y_{i}\\}_{i=1}^{N}\subset M$, $\phi\in\emph{Lip}(M)$ and
$\nu=\sum_{i=1}^{N}\lambda_{i}\delta_{y_{i}}$, where $\lambda_{i}\in[0,1]$ and
$\sum_{i=1}^{N}\lambda_{i}=1$, $N\in\mathbb{N}$. Then
$\mu=\sum_{i=1}^{N}\lambda_{i}\mu_{i}$ solves (K-), where
$\\{\mu_{i}\\}_{i=1}^{N}\subset\mathscr{P}(M)$, and each $\mu_{i}$ satisfies
(4.4) for $y_{0}=y_{i}$. Moreover
$\mathrm{supp}\,(\mu)\subset\cup_{i=1}^{N}\\{x:y_{i}\in\partial_{c}\phi(x)\\}$.

###### Proof.

We first suppose $\mu=\sum_{i=1}^{N}\lambda_{i}\mu_{i}$. Consider the formula
(K-) in the case of $\nu_{i}=\delta_{y_{i}}$, $i=1,2,\cdots,N$:

$\displaystyle\int_{M}T^{-}\phi(y)\,d\nu_{i}=\inf_{\pi_{i}\in\Gamma\,(\mu_{i},\nu_{i})}\int_{M\times
M}\phi(x)+c(x,y)\,d\pi_{i}.$

It holds true if and only if
$\mathrm{supp}\,\,(\mu_{i})\subset\\{x:y_{i}\in\partial_{c}\phi(x)\\}$.
Proposition 4.7 guarantees the existence of such $\mu_{i}$. For
$\pi_{i}\in\Gamma_{o}(\mu_{i},\delta_{y_{i}})$,

$\displaystyle\int_{M}T^{-}\phi(y)\,d\nu$
$\displaystyle=\sum_{i=1}^{N}\lambda_{i}\int_{M}T^{-}\phi(y)\,d\delta_{y_{i}}$
$\displaystyle=\sum_{i=1}^{N}\lambda_{i}\int_{M\times
M}\phi(x)+c(x,y)\,d\pi_{i}$ $\displaystyle=\int_{M\times
M}\phi(x)+c(x,y)\,d\pi,$

where $\pi:=\sum_{i=1}^{N}\lambda_{i}\pi_{i}$. Thus
$\pi\in\Gamma_{o}(\mu,\nu)$ and $\mu$ satisfies (K-). For the rest of the
proof, we only need to recall that
$\mathrm{supp}\,\,(\mu_{1}+\mu_{2})=\mathrm{supp}\,\,(\mu_{1})\cup\mathrm{supp}\,\,(\mu_{2})$
for arbitrary $\mu_{1},\mu_{2}\in\mathscr{P}\,(M)$. ∎

######  Corollary 4.9.

Under the assumptions in Lemma 4.8, we have

  1. (1)

for any $\Pi\in\mathscr{P}(T^{*}M)$, with
$\mathrm{supp}\,(\Pi)\subset\cup_{i=1}^{N}(y_{i},D^{*}T^{-}\phi(y_{i}))$ and
$(p_{x})_{\\#}\Pi=\nu$, there exists
$\mu_{\Pi}:=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\Pi$, which holds (K-);

  2. (2)

for any $\mu\in\mathscr{P}(M)$ satisfies (K-), there exists
$\Pi_{\mu}\in\mathscr{P}(T^{*}M)$ with $(p_{x})_{\\#}\Pi_{\mu}=\nu$ and
$\mathrm{supp}\,(\Pi_{\mu})\subset\cup_{i=1}^{N}(y_{i},D^{+}T^{-}\phi(y_{i}))$,
such that $\mu=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\Pi_{\mu}$.

###### Proof.

By disintegration theorem, there exist measures $\\{\Pi_{y_{i}}\\}_{i=1}^{N}$
with $\Pi=\sum_{i=1}^{N}\lambda_{i}\Pi_{y_{i}}\times\delta_{y_{i}}$. As a
direct consequence of Lemma 4.8 there exist $\mu_{i}$ satisfies (4.4), then
$\mu_{\Pi}:=\sum_{i=1}^{N}\mu_{i}$ solves (K-).

Conversely, if $\mu\in\mathscr{P}(M)$ satisfies (K-), due to Proposition 4.7,
there exists $\rho_{i}\in\mathscr{P}(\mathbb{R}^{d})$ such that, for each $i$,
the support of which contains in $D^{+}T^{-}\phi(y_{i})$,
$\mu=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}(\delta_{y_{i}}\times\rho_{i})$. Defining
$\Pi_{\mu}:=\sum_{i=1}^{N}\lambda_{i}\delta_{y_{i}}\times\rho_{i}$ we obtain

$\displaystyle(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\Pi_{\mu}=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\left(\sum_{i=1}^{N}\lambda_{i}\delta_{y_{i}}\times\rho_{i}\right)=\sum_{i=1}^{N}\lambda_{i}(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\left(\delta_{y_{i}}\times\rho_{i}\right)=\mu.$

In addition, it is easy to check that
$\mathrm{supp}\,(\Pi_{\mu})\subset\cup_{i=1}^{N}(y_{i},D^{+}T^{-}\phi(y_{i}))$
and $(p_{x})_{\\#}\Pi_{\mu}=\nu$. ∎

######  Lemma 4.10.

Let $(X,d)$ be a metric space,
$\\{\mu_{n},\mu\\}_{n=1}^{\infty}\subset\mathscr{P}(X)$.

  1. (1)

_(Portmanteau)_
$\mu_{n}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\mu$
if and only if $\liminf_{n\to\infty}\mu_{n}(G)\geqslant\mu(G)$ for any open
subset $G\subset X$;

  2. (2)

$\mathrm{supp}\,\,(\mu)\subset\mathop{\lim\inf}_{n\to\infty}\mathrm{supp}\,(\mu_{n})$,
if
$\mu_{n}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\mu$.

###### Proof.

The reader can refer [11, Theorem 2.1] for the proof of (1). (2) is a direct
consequence of (1). Actually, for any $x\in\mathrm{supp}\,\,(\mu)$ and any
neighborhood $U_{x}$ of $x$ in $X$ with $\mu\,(U_{x})>0$. Portmanteau’s
theorem implies that for all sequences
$\mu_{n}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\mu$,

$\displaystyle\mathop{\lim\inf}_{n\to\infty}\mu_{n}(U_{x})\geqslant\mu\,(U_{x})>0.$

By the definition of limit inferior of sets,
$x\in\mathop{\lim\inf}_{n\to\infty}\mathrm{supp}\,\,(\mu_{n})$. ∎

######  Theorem 4.11.

Suppose $\nu\in\mathscr{P}(M)$ and $\phi\in\emph{Lip}\,(M)$. Then there exists
some $\mu\in\mathscr{P}(M)$ solving (K-) and satisfying

$\displaystyle\mathrm{supp}\,(\mu)\subset\mathop{\lim\inf}_{k\to\infty}\left\\{x\in
M:\partial_{c}\phi(x)\cap C_{k}\neq\varnothing\right\\},$

for some $a_{k}$-net $C_{k}$ of $\mathrm{supp}\,\,(\nu)$, where $a_{k}\searrow
0$ as $k\to\infty$.

###### Proof.

Because of the closedness of $\mathrm{supp}\,\,(\nu)$ for
$\nu\in\mathscr{P}(M)$ and the compactness of $M$, $\mathrm{supp}\,(\nu)$ is a
compact subset of $M$. Thus $\mathrm{supp}\,(\nu)$ has a finite $1/n$-net
$C_{n}:=\\{y_{i}^{(n)}\\}_{i=1}^{N_{n}}$ for each $n\in\mathbb{N}^{*}$. Now we
have $\mathrm{supp}\,(\nu)\subset\cup_{i=1}^{N_{n}}B(y_{i}^{(n)},1/n)$.

Choose $\\{r_{i}\\}_{i=1}^{N_{n}}\subset\mathbb{Q}^{+}$ satisfying
$\sum_{i=1}^{N_{n}}r_{i}=1$ and
$\sum_{i=1}^{N_{n}}\left|r_{i}-\nu\,(B(y_{i}^{(n)},1/n))\right|\leqslant 1/n$.
Let $\nu_{n}:=\sum_{n=1}^{N_{n}}r_{i}\delta_{y_{i}^{(n)}}$. According to the
argument in [11, Section 6],
$\nu_{n}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\nu$,
i.e.,

$\displaystyle\int_{M}T^{-}\phi(y)\,d\nu=\lim_{n\to\infty}\int_{M}T^{-}\phi(y)\,d\nu_{n}.$

By Proposition 4.8, we can find some $\mu_{n}\in\mathscr{P}(M)$ and
$\pi_{n}\in\Gamma_{o}(\mu_{n},\nu_{n})$, such that

$\displaystyle\int_{M}T^{-}\phi(y)\,d\nu_{n}=\int_{M\times
M}\phi(x)+c(x,y)\,d\pi_{n}.$

and

$\mathrm{supp}\,(\mu_{n})\subset\bigcup_{i=1}^{N_{n}}\left\\{x:y_{i}^{(n)}\in\partial_{c}\phi(x)\right\\}=\\{x:\partial_{c}\phi(x)\cap
C_{n}\neq\varnothing\\}.$ (4.6)

Note that the sequence $\\{\mu_{n}\\}_{n=1}^{\infty}$ is tight on
$\mathscr{P}(M)$. Invoking Prokhorov theorem, we can get a convergent
subsequence of $\\{\mu_{n_{k}}\\}_{k=1}^{\infty}$, such that
$\mu_{n_{k}}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\mu\in\mathscr{P}(M)$.
The stability of the transport plan implies there exists some subsequence
$\\{\pi_{n_{k}}\\}_{k=1}^{\infty}$ and
$\pi_{n_{k}}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\pi$
with $\pi\in\Gamma_{o}(\mu,\nu)$. Thus,

$\displaystyle\int_{M}T^{-}\phi(y)\,d\nu$
$\displaystyle=\lim_{k\to\infty}\int_{M}T^{-}\phi(y)\,d\nu_{n_{k}}$
$\displaystyle=\lim_{k\to\infty}\int_{M\times M}\phi(x)+c(x,y)\,d\pi_{n_{k}}$
$\displaystyle=\int_{M\times M}\phi(x)+c(x,y)\,d\pi,$

i.e., $\mu$ solves (K-). By Lemma 4.10 and (4.6) we have

$\displaystyle\mathrm{supp}\,(\mu)\subset\mathop{\lim\inf}_{k\to\infty}\mathrm{supp}\,\,(\mu_{n_{k}})\subset\mathop{\lim\inf}_{k\to\infty}\\{x:\partial_{c}\phi(x)\cap
C_{n_{k}}\neq\varnothing\\}.$

This completes the proof by choosing $a_{k}=1/n_{k}$. ∎

######  Corollary 4.12.

Under the assumptions in Theorem 4.11, there exists $\mu\in\mathscr{P}(M)$
solving (K-) and satisfying

$\displaystyle\mathrm{supp}\,(\mu)\subset
p_{x}\circ\Phi_{H}^{t,0}\,(\mathrm{graph}\,(D^{*}T^{-}\phi(\mathrm{supp}\,(\nu)))).$

###### Proof.

Following the notations in the proof of Theorem 4.11, for each $\nu_{n}$ we
can pick some $\Pi_{n}\in\mathscr{P}(T^{*}M)$ whose support is contained in
$\bigcup_{i=1}^{N_{n}}(y_{i}^{(n)},D^{*}T^{-}\phi(y_{i}^{(n)}))$.
$\mu_{n}:=(p_{x}\circ\Phi_{H}^{t,0})_{\\#}\Pi_{n}$ solves (K-) by Corollary
4.9. Since
$\nu_{n}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\nu$
and $\\{\Pi_{n}\\}$ is tight, $\\{\Pi_{n}\\}$ admits a subsequence
$\\{\Pi_{n_{k}}\\}$ converging to some $\Pi\in\mathscr{P}(T^{*}M)$. In
addition,
$\mu_{n_{k}}\mathrel{\ensurestackMath{\stackon[1pt]{\rightharpoonup}{\scriptstyle\ast}}}\mu$
with $\mu$ solving (K-) and $(p_{x})_{\\#}\Pi=\nu$. Consequently, by Lemma
4.10,

$\displaystyle\mathrm{supp}\,(\Pi)\subset\liminf_{k\to\infty}\cup_{i=1}^{N_{n_{k}}}\left(y_{i},D^{*}T^{-}\phi\left(y_{i}^{(n_{k})}\right)\right)\subset\mathrm{graph}\,(D^{*}T^{-}\phi(\mathrm{supp}\,(\nu))).$

Combining with Lemma 4.3 and the fact that $\Phi_{H}^{0,t}$ is a
diffeomorphism, we get

$\displaystyle\mathrm{supp}\,(\mu)$
$\displaystyle=\mathrm{supp}\,((p_{x}\circ\Phi_{H}^{t,0})_{\\#}\Pi)$
$\displaystyle=p_{x}\circ\Phi_{H}^{t,0}(\mathrm{supp}\,(\Pi))$
$\displaystyle\subset
p_{x}\circ\Phi_{H}^{t,0}\,(\mathrm{graph}\,(D^{*}T^{-}\phi(\mathrm{supp}\,(\nu)))).$

This completes the proof. ∎

Finally, we discuss the Monge problem. We suppose $X$ and $Y$ are both Polish
spaces.

######  Proposition 4.13.

Assume that $\mu\in\mathscr{P}(X)$, $\nu\in\mathscr{P}(Y)$ and $\mu$ has no
atom. $T:X\to Y$, $T_{\\#}\mu=\nu$. Then the followings are equivalent.

  1. (1)

$T$ is an optimal transport map;

  2. (2)

There exists $\phi\in\emph{Lip}_{b}(X)$, such that
$\int_{X}T^{-}\phi(T(x))\,d\mu=\int_{X}\phi(x)+c(x,T(x))\,d\mu$;

  3. (3)

There exists $\phi\in\emph{Lip}_{b}(X)$, such that
$T(x)\in\partial_{c}\phi(x)$ holds for $\mu$-a.e. $x\in X$.

###### Proof.

$(1)\Rightarrow(2)$. Since $\mu$ has no atom, according to Pratelli theorem
(See [3, Theorem 2.2]), if $T$ is an optimal map,

$\displaystyle\int_{X}c(x,T(x))\,d\mu=\inf_{\pi\in\Gamma\,(\mu,\nu)}\int_{X\times
Y}c(x,y)\,d\pi.$

Thus for any $\phi\in\text{Lip}_{b}(X)$, we have

$\displaystyle\int_{X}\phi(x)+c(x,T(x))\,d\mu=\inf_{\pi\in\Gamma\,(\mu,\nu)}\int_{X\times
Y}\phi(x)+c(x,y)\,d\pi.$

Choose some $\phi$ solving (K-), then

$\displaystyle\int_{X}\phi(x)+c(x,T(x))\,d\mu=\int_{Y}T^{-}\phi(y)\,d\nu=\int_{X}T^{-}\phi(T(x))\,d\mu,$

with $T_{\\#}\mu=\nu$.

$(2)\Rightarrow(3)$. We only need to observe that for any $x\in X$,
$T^{-}\phi(T(x))\leqslant\phi(x)+c(x,T(x))$. By the equality in (2),
$x=\arg\min\\{\phi(\cdot)+c(\cdot,T(x))\\}$ for $\mu$-a.e. $x\in X$.

$(3)\Rightarrow(1)$. Given $\phi\in\text{Lip}_{b}(X)$. We have

$\displaystyle\int_{X}T^{-}\phi(T(x))\,d\mu$
$\displaystyle=\int_{Y}T^{-}\phi(y)\,d\nu$
$\displaystyle\leqslant\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times
Y}\phi(x)+c(x,y)\,d\pi$ $\displaystyle\leqslant\int_{X\times
Y}\phi(x)+c(x,y)\,d(\text{id}\times T)_{\\#}\mu$
$\displaystyle=\int_{X}\phi(x)+c(x,T(x))\,d\mu.$

If (3) holds, then

$\int_{X}T^{-}\phi(T(x))\,d\mu=\int_{X}\phi(x)+c(x,T(x))\,d\mu=\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times
Y}\phi(x)+c(x,y)\,d\pi.$ (4.7)

It is well known that the infimum of Monge’s problem is not less than the
minimum of the associated Kantorovich’s problem, thus $T$ in (4.7) is the
optimal transport map. ∎

######  Remark 4.14.

The function $\phi$ in (2) or (3) of Proposition 4.13 satisfies (K-).

###  4.3. Random Lax-Oleinik operators

Suppose $(X,d)$ is a metric space. Let $(\mathscr{P}_{p}(X),W_{p})$ be the
$p$-Wasserstein space for $p\in[1,+\infty)$, where $W_{p}$ is the $p$-order
Wasserstein metric, defined as

$\displaystyle
W_{p}(\mu,\nu):=\left\\{\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{X^{2}}d^{p}(x,y)\,d\gamma\right\\}^{\frac{1}{p}}.$

If $X$ is a Polish space, then so is $(\mathscr{P}_{p}(X),W_{p})$.

Set $\kappa_{1},\kappa_{2}\geqslant 0$. A function
$\phi:X\to[-\infty,+\infty]$ is said to be $(\kappa_{1},\kappa_{2})$-Lipschitz
in the large, if for any $x,y\in X$, we have

$\displaystyle|\phi(x)-\phi(y)|\leqslant\kappa_{1}d(x,y)+\kappa_{2}.$

######  Remark 4.15.

For any function $\phi:X\to\mathbb{R}$, we have:

  1. (1)

$\phi\in\mathrm{Lip\,}(X)$ implies $\phi$ is
$(\mathrm{Lip\,}(\phi),0)$-Lipschitz in the large;

  2. (2)

if $X$ is compact, $\phi$ is $(\kappa_{1},\kappa_{2})$-Lipschitz in the large
if and only if $\phi$ is bounded;

  3. (3)

if $X$ is a geodesic space, $\phi\in\mathrm{UC}\,(X)$ if and only for any
$\varepsilon>0$, there exists $K_{\varepsilon}>0$, such that $\phi$ is
$(\varepsilon,K_{\varepsilon})$-Lipschitz in the large.

######  Lemma 4.16.

Assume that $\phi$ is $(\kappa_{1},\kappa_{2})$-Lipchitz in the large,
$\mu\in\mathscr{P}_{1}(\mathbb{R}^{d})$. Then $\phi\in L^{1}(\mu)$.
Particularly, if $\phi\in\mathrm{UC}\,(\mathbb{R}^{d})$ and
$\mu\in\mathscr{P}_{1}(\mathbb{R}^{d})$, then $\phi\in L^{1}(\mu)$.

###### Proof.

The proof is directly from the definition of Lipschitz in the large. Indeed,
for arbitrary $x_{0}\in X$,

$\displaystyle\int_{\mathbb{R}^{d}}|\phi(x)-\phi(x_{0})|\,d\mu\leqslant\int_{\mathbb{R}^{d}}(\kappa_{1}d(x,x_{0})+\kappa_{2})\,d\mu<+\infty.$

Thus, this completes the proof with Remark 4.15. ∎

######  Definition 4.17 ([4, 5], Potential energy).

Given $\phi\in L^{0}(X;(-\infty,+\infty])$ which is bounded from below.
Associated with $\phi$ we define a functional of $\mathscr{P}_{p}(X)$, called
the potential energy associated to $\phi$,

$\displaystyle\Phi(\mu):=\int_{X}\phi\,d\mu,\qquad\mu\in\mathscr{P}_{p}(X).$

We denote this functional by $\phi(\mu)$ for short without getting confused.

######  Definition 4.18 (Dynamical cost functional).

Let $s,t\in\mathbb{R}$ with $s<t$, $\mu\in\mathscr{P}(X)$ and
$\nu\in\mathscr{P}(Y)$. The dynamical cost functional associated to $c^{s,t}$
is

$\displaystyle C^{s,t}(\mu,\nu):=\inf_{\pi\in\Gamma(\mu,\nu)}\int_{X\times
Y}c^{s,t}(x,y)\,d\pi=\inf_{\begin{subarray}{c}\mathrm{law}(x)=\mu\\\
\mathrm{law}(y)=\nu\end{subarray}}\mathbb{E}(c^{s,t}(x,y)),$

where $x(\omega)$ and $y(\omega)$ in the last term are $X$-valued and
$Y$-valued random variables of some probability space
$(\Omega,\mathscr{F},\mathbb{P})$ respectively.

We remark that Brenier [12] and Knott-Smith [24] proved the existence of a
probability space $(\Omega,\mathscr{F},\mathbb{P})$ such that for arbitrary
$\mu\in\mathscr{P}_{2}(\mathbb{R}^{d})$, we can always find a random variable
$T\in L^{0}(\Omega;\mathbb{R}^{d})$, which satisfies $\mathrm{law\,}(T)=\mu$.
Indeed, we choose $\Omega=\mathbb{R}^{d}$, $\mathscr{F}$ is the set of
Lebesgue measurable subsets of $\mathbb{R}^{d}$ and
$\mathbb{P}=\mathscr{L}^{d}|_{B(0,R)}$, where $R>0$ makes
$\mathscr{L}^{d}(B(0,R))=1$. If $c(x,y)=\frac{1}{2}|x-y|^{2}$, for the
corresponding Monge’s transport problem, there exists an optimal map $T\in
L^{0}(\mathbb{R}^{d},\mathbb{R}^{d})$ of $\mathcal{C}(\mathbb{P},\mu)$ such
that $T_{\\#}\mathbb{P}=\mu$, i.e., $\mathrm{law\,}(T)=\mu$. By Theorem 10.28
and Example 10.36 of [27], the argument for the case $X=Y=M$ with $M$ compact
manifold and $\mu\in\mathscr{P}(M)$ is similar.

Let us recall some classical results on measurable selection in the field of
measure theory, infinite dimensional analysis and optimal transport.

######  Lemma 4.19 ([2, Theorem 18.19]).

Let $X$ be a separable metrizable space and $(S,\Sigma)$ a measurable space.
Let $\varphi:S\leadsto X$ be a weakly measurable (see [2, Section 17])
nonempty compact set-valued map, and suppose $F:S\times X\to\mathbb{R}$ is a
Carathéodory function. Define the marginal functions $f:S\to\mathbb{R}$ by

$\displaystyle f(s):=\max_{x\in\varphi(s)}F(s,x),$

and the correspondence $\Lambda:S\leadsto X$ of maximizers by
$\Lambda(s):=\mathop{\arg\max}_{x\in\varphi(s)}\\{F(s,x)\\}$, then

  1. (1)

$f$ is measurable;

  2. (2)

$\Lambda(s)$ is a nonempty compact set for each $s\in S$ and exists a
measurable selection $\lambda(s)$.

######  Lemma 4.20 ([27, Corollary 5.22]).

Let $X$ and $Y$ be Polish spaces and let $c:X\times Y\to\mathbb{R}$ be a
continuous cost function with a lower bound. Suppose the map

$\displaystyle\Omega$ $\displaystyle\to\mathscr{P}(X)\times\mathscr{P}(Y)$
$\displaystyle\omega$ $\displaystyle\mapsto(\mu_{\omega},\nu_{\omega})$

is measurable. Then there is a measurable selection
$\omega\mapsto\pi_{\omega}$ such that for each $\omega$, $\pi_{\omega}$ is an
optimal plan between $\mu_{\omega}$ and $\nu_{\omega}$.

In the sequel, we consider the case of $X=Y=M$, a connected and closed
manifold of dimension $d$. and $c^{s,t}(x,y):=h(s,t,x,y)$, the fundamental
solution from weak KAM theory.

######  Definition 4.21 (Random Lax-Oleinik operator).

For any function $\phi:M\to[-\infty,+\infty]$, $t>0$, $\mu\in\mathscr{P}(M)$,
we define

$\displaystyle P_{t}^{-}\phi(\mu)$
$\displaystyle:=\inf_{\nu\in\mathscr{P}(M)}\\{\phi(\nu)+C^{0,t}(\nu,\mu)\\},$
$\displaystyle P_{t}^{+}\phi(\mu)$
$\displaystyle:=\sup_{\nu\in\mathscr{P}(M)}\\{\phi(\nu)-C^{0,t}(\mu,\nu)\\}.$

######  Proposition 4.22 ([27, Theorem 7.21]).

Let $s,t\in\mathbb{R}$ and $\mu,\nu\in\mathscr{P}(M)$ such that
$C^{s,t}(\mu,\nu)$ is finite. Then

$\displaystyle C^{s,t}(\mu,\nu)$
$\displaystyle=\min_{\pi\in\Gamma(\mu,\nu)}\int_{M^{2}}h(s,t,x,y)\,d\pi=\min_{\xi\in
L_{\mu,\nu}^{s,t}}\int_{\Omega}\int_{s}^{t}L(\xi(r,\omega),\dot{\xi}(r,\omega))\,drd\mathbb{P},$

where $L_{\mu,\nu}^{s,t}:=\\{\xi\in
L^{0}(\Omega;AC([s,t];M)):\mathrm{law}\,(\xi(s,\cdot))=\mu,\mathrm{law}\,(\xi(t,\cdot))=\nu\\}$.

######  Theorem 4.23.

Suppose $\phi\in C(M)$ and $t>0$. Then, for any $\nu\in\mathscr{P}(M)$ the
operator $P_{t}^{-}\phi(\nu)$ is finite-valued. There exist
$\mu\in\mathscr{P}(M)$ and $\xi\in L_{\mu,\nu}^{0,t}$ such that

$\displaystyle P_{t}^{-}\phi(\nu)$ $\displaystyle=\phi(\mu)+C^{0,t}(\mu,\nu)$
$\displaystyle=\int_{\Omega}\phi(\xi(0,\omega))+\int_{0}^{t}L(\xi(s,\omega),\dot{\xi}(s,\omega))\,dsd\mathbb{P}$
$\displaystyle=\int_{0}^{t}\int_{TM}\phi(x)+L(x,v)\,d\tilde{\mu}_{s}ds,$

where
$\tilde{\mu}_{s}:=\mathrm{law\,}(\xi(s,\cdot),\dot{\xi}(s,\cdot))\in\mathscr{P}(TM)$
is determined by the associated Euler-Lagrangian flow.

Moreover, $P_{t}^{-}\phi(\nu)=T_{t}^{-}\phi(\nu)$ for any
$\nu\in\mathscr{P}(M)$, i.e., for any $\nu\in\mathscr{P}(M)$, there exists
$\mu\in\mathscr{P}(M)$ solving (K-) .

###### Proof.

It is worth noting that for arbitrary $x,y\in M$,
$T_{t}^{-}\phi(y)\leqslant\phi(x)+h(0,t,x,y)$. Thus, given
$\nu\in\mathscr{P}(M)$, for all $\mu\in\mathscr{P}(M)$ and
$\pi\in\Gamma(\mu,\nu)$,

$\displaystyle\int_{M}T_{t}^{-}\phi(y)\,d\nu=\int_{M\times
M}T_{t}^{-}\phi(y)\,d\pi\leqslant\int_{M\times M}\phi(x)+h(0,t,x,y)\,d\pi,$

which means

$T_{t}^{-}\phi(\nu)=\int_{M}T_{t}^{-}\phi(y)\,d\nu\leqslant\inf_{\mu\in\mathscr{P}(M)}\inf_{\pi\in\Gamma(\mu,\nu)}\int_{M\times
M}\phi(x)+h(0,t,x,y)\,d\pi=P_{t}^{-}\phi(\nu).$ (4.8)

On the other hand, we can find a $M$-valued random variable $y(\omega)\in
L^{0}(\Omega;M)$ of $(\Omega,\mathscr{F},\mathbb{P})$ with
$\mathrm{law\,}(y)=\nu$. Recall that, due to a priori estimates on the
minimizer (for example, see [18, Lemma 3.1]), the marginal function can be
rewritten as

$\displaystyle T_{t}^{-}\phi(y(\omega))=\min_{x\in B(y(\omega),\lambda
t)}\\{\phi(x)+h(0,t,x,y(\omega))\\},\qquad\forall\omega\in\Omega,$

where $\lambda>0$ only depends on the Lagrangian and Lipschitz in the large
constant of $\phi$. It is easy to check that $\omega\mapsto
B(y(\omega),\lambda t)$ is a measurable and set-valued map with compact value.
According to Lemma 4.19, it admits a measurable selection

$\displaystyle\omega\mapsto x(\omega)\in\mathop{\arg\min}_{x\in
B(y(\omega),\lambda t)}\\{\phi(x)+h(0,t,x,y(\omega))\\}\subset
B(y(\omega),\lambda t).$

Therefore, define $\mu:=x_{\\#}\mathbb{P}$, then $\mu\in\mathscr{P}(M)$ and

$\displaystyle\int_{M}T_{t}^{-}\phi(y)\,d\nu$
$\displaystyle=\int_{\Omega}T_{t}^{-}\phi(y(\omega))\,d\mathbb{P}$
$\displaystyle=\int_{\Omega}\phi(x(\omega))+h(0,t,x(\omega),y(\omega))\,d\mathbb{P}$
$\displaystyle=\int_{M\times
M}\phi(x)+h(0,t,x,y)\,d(x,y)_{\\#}\mathbb{P}\geqslant\phi(\mu)+C^{0,t}(\mu,\nu).$

Since $\omega\mapsto(x(\omega),y(\omega))$ is measurable, invoking Lemma 4.20
and the existence of random curves, we can find some
$\xi(\omega)\in\Gamma_{x(\omega),y(\omega)}^{0,t}$, which satisfies

$\displaystyle
T_{t}^{-}\phi(y(\omega))=\phi(x(\omega))+\int_{0}^{t}L(\xi(s,\omega),\dot{\xi}(s,\omega))\,ds$

and $\xi\in L^{0}(\Omega;\mathrm{Lip\,}([0,t];M))$. As a consequence we obtain

$\displaystyle\int_{\mathbb{R}^{d}}T_{t}^{-}\phi\,d\nu\geqslant\phi(\mu)+C^{0,t}(\mu,\nu)\geqslant
P_{t}^{-}\phi(\nu).$

Together with (4.8),

$\displaystyle\int_{\mathbb{R}^{d}}T_{t}^{-}\phi\,d\nu=\phi(\mu)+C^{0,t}(\mu,\nu)=P_{t}^{-}\phi(\nu)=\int_{\Omega}\phi(\xi(0,\omega))+\int_{0}^{t}L(\xi(s,\omega),\dot{\xi}(s,\omega))\,dsd\mathbb{P}.$

Finally we remark that if $L(x,\cdot)$ is strictly convex, then
$\xi(\cdot,\omega)\in C^{2}([0,t];M)$ for any $\omega\in\Omega$, and
$(\xi(\cdot,\omega),\dot{\xi}(\cdot,\omega))\in TM$ solves the Euler-Lagrange
equation. ∎

An similar reasoning of the proof of Theorem 4.23 leads to

######  Theorem 4.24.

Suppose $\phi\in C(M)$, $t>0$. Then, for any $\mu\in\mathscr{P}(M)$,
$P_{t}^{+}\phi(\mu)$ is finite-valued and there exists $\nu\in\mathscr{P}(M)$
and $\xi\in L_{\mu,\nu}^{0,t}$ such that

$\displaystyle P_{t}^{+}\phi(\nu)$ $\displaystyle=\phi(\nu)-C^{0,t}(\mu,\nu)$
$\displaystyle=\int_{\Omega}\phi(\xi(t,\omega))-\int_{0}^{t}L(\xi(s,\omega),\dot{\xi}(s,\omega))\,dsd\mathbb{P}$
$\displaystyle=\int_{0}^{t}\int_{TM}\phi(x)-L(x,v)\,d\tilde{\nu}_{s}ds,$

where
$\tilde{\nu}_{s}:=\mathrm{law\,}(\xi(s,\cdot),\dot{\xi}(s,\cdot))\in\mathscr{P}(TM)$
is determined by the associated Euler-Lagrangian flow.

Moreover, $P_{t}^{+}\phi(\mu)=T_{t}^{+}\phi(\mu)$ for any
$\mu\in\mathscr{P}(M)$, i.e., for any $\mu\in\mathscr{P}(M)$ there exists
$\nu\in\mathscr{P}(M)$ solving (K+).

##  Appendix A Proofs of some Lemmata

###### Proof of Lemma 3.13.

By some basic results from the theory of semiconcave functions, for any $s\in
S$, $x\in\mathbb{R}^{d}$ and $p\in D^{+}_{x}F(s,x)$, define
$\phi(s,x,\cdot)=F(t,x)+\langle p,\cdot-x\rangle+\frac{C}{2}|\cdot-x|^{2}$.
Then, $\phi(s,x,\cdot)$ touches $F(s,\cdot)$ from above at $x$ and
$p=D\phi(s,x,x)$. Thus

$\displaystyle F(s,x)=\inf_{x\in\mathbb{R}^{d}}\phi(s,x,x),\qquad
u(x)=\inf_{s\in S,x\in\mathbb{R}^{d}}\phi(s,x,x).$

Then, our conclusion is a direct consequence of the property of a standard
marginal function (see, for instance, [19, Theorem 3.4.4]). ∎

###### Proof of Lemma 4.3.

Let $x_{0}\in p_{x}(\mathrm{supp}\,\,(\pi))$, then there exists $y_{0}\in Y$,
$(x_{0},y_{0})\in\mathrm{supp}\,\,(\pi)$, i.e. for any neighborhood
$U_{x_{0},y_{0}}$ of $(x_{0},y_{0})$ in $X\times Y$,
$\pi\,(U_{x_{0},y_{0}})>0$. Let $U_{x_{0}}$ be any neighborhood of $x_{0}\in
X$. Since $U_{x_{0}}\times Y$ is also a neighborhood of $(x_{0},y_{0})$, we
have

$(p_{x})_{\\#}\pi\,(U_{x_{0}})=\pi\,(p_{x}^{-1}(U_{x_{0}}))=\pi\,(U_{x_{0}}\times
Y)>0,$ (A.1)

which implies $x_{0}\in\mathrm{supp}\,((p_{x})_{\\#}\pi)$.

Now, we turn to the proof of (2). We only need to prove the inclusion
$\mathrm{supp}\,((p_{x})_{\\#}\pi)\subset p_{x}(\mathrm{supp}\,(\pi))$. Let
$x_{0}\in\mathrm{supp}\,((p_{x})_{\\#}\pi)$, then (A.1) holds true for any
neighborhood $U_{x_{0}}$ of $x_{0}\in X$. Thus, it suffices to prove that
there exists $y_{0}\in Y$ such that $\pi\,(U_{x_{0},y_{0}})>0$ for any
$U_{x_{0},y_{0}}\subset X\times Y$. We prove by contradiction. Assume that for
any $y\in Y$ there exists an open set $V_{x_{0},y}$ containing $(x_{0},y)$
such that $\pi\,(V_{x_{0},y})=0$. More precisely, for each $y$, there exist
$R_{y}^{X},R_{y}^{Y}>0$ with $B(x_{0},R_{y}^{X})\times B(y,R_{y}^{Y})\subset
V_{x_{0},y}$, and $\pi\,(B(x_{0},R_{y}^{X})\times B(y,R_{y}^{Y}))=0$. Due to
the compactness of $Y$, the open cover $\\{B(x_{0},R_{y}^{X})\times
B(y,R_{y}^{Y})\\}_{y\in Y}$ of $\\{x_{0}\\}\times Y$ admits a finite subcover
$\\{B(x_{0},R_{y_{i}}^{X})\times B(y_{i},R_{y_{i}}^{Y})\\}_{i=1}^{N}$. Set
$R_{x_{0}}:=\min_{1\leqslant i\leqslant
N}\\{R_{y_{i}}^{X},R_{y_{i}}^{Y}\\}>0$. Then

$\displaystyle\\{x_{0}\\}\times Y\subset B(x_{0},R_{x_{0}})\times
Y\subset\bigcup_{i=1}^{N}\left\\{B(x_{0},R_{y_{i}}^{X})\times
B(y_{i},R_{y_{i}}^{Y})\right\\}.$

It follows that

$\displaystyle\pi\,(B(x_{0},R_{x_{0}})\times Y)$
$\displaystyle\leqslant\pi\left(\bigcup_{i=1}^{N}\left\\{B(x_{0},R_{y_{i}}^{X})\times
B(y_{i},R_{y_{i}}^{Y})\right\\}\right)$
$\displaystyle\leqslant\sum_{i=1}^{N}\pi\left(B(x_{0},R_{y_{i}}^{X})\times
B(y_{i},R_{y_{i}}^{Y})\right)=0.$

Therefore, $B(x_{0},R_{x_{0}})$, as a neighborhood of $x_{0}$, does not
satisfy (A.1). This leads to a contradiction. ∎

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