A duality approach to a price formation MFG model

Yuri Ashrafyan , Tigran Bakaryan , Diogo Gomes and Julian Gutierrez

Abstract.

We study the connection between the Aubry-Mather theory and a mean-field game (MFG) price-formation model. We introduce a framework for Mather measures that is suited for constrained time-dependent problems in ${\mathbb{R}}$ . Then, we propose a variational problem on a space of measures, from which we obtain a duality relation involving the MFG problem examined in [35].

Key words and phrases:

Mean Field Games; Price formation; Common noise; ADD Keywords

King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900. Saudi Arabia. e-mail: [email protected]

The authors were partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452.

1. introduction

This paper studies the connection between Aubry-Mather theory and certain mean-field games (MFG) that model price formation. More precisely, we consider the MFG system

\begin{cases}-u_{t}(t,x)+H(x,\varpi(t)+u_{x}(t,x))=0&\quad(t,x)\in[0,T]\times{\mathbb{R}}\\ m_{t}(t,x)-\big{(}H_{p}(x,\varpi(t)+u_{x}(t,x))m(t,x)\big{)}_{x}=0&\quad(t,x)\in[0,T]\times{\mathbb{R}}\\ -\int_{{\mathbb{R}}}H_{p}(x,\varpi(t)+u_{x}(t,x))m(t,x)dx=Q(t)&\quad t\in[0,T]\end{cases},

(1.1)

subject to initial-terminal conditions

\begin{cases}u(T,x)=u_{T}(x)\\ m(0,x)=m_{0}(x)\end{cases}\quad x\in{\mathbb{R}},

(1.2)

where $Q$ , $u_{T}$ , and $m_{0}$ are given functions, $m_{0}$ is a probability measure on ${\mathbb{R}}$ , and the triplet $(u,m,\varpi)$ is the unknown. Here, the state of a typical agent is the variable $x\in{\mathbb{R}}$ and represents the assets of that agent. The distribution of assets in the population of the agents at time $t$ is encoded in the probability measure $m(\cdot,t)$ . The agents change their assets by trading at a price $\varpi(t)$ . The trading is subject to a balance condition encoded in the third equation in (1.1). This integral constraint that guarantees supply $Q(t)$ meets demand is represented by the term on the left-hand side of that condition.

As introduced in [35], $u$ is the value function of an agent who trades a commodity with supply $Q$ and price $\varpi$ . The function $u$ is characterized by the first equation in (1.1) and the terminal condition in (1.2). Each agent selects their trading rate in order to minimize a given cost functional (see (1.7) below). The optimal control selection is $-H_{p}(x,\varpi(t)+u_{x}(t,x))$ . Under this optimal control, the density $m$ describing the population of agents evolves according to the second equation in (1.1) and the initial condition in (1.2). The third equation in (1.1), which we refer to as the balance condition, is an integral constraint that guarantees supply meets demand.

Remark 1.1.

The notion of solutions of (1.1) and (1.2) we consider is the following: $u\in C([0,T]\times{\mathbb{R}})$ solves the first equation in the viscosity sense, $m\in C([0,T],{\mathcal{P}}({\mathbb{R}}))$ solves the second equation in the distributional sense, and $\varpi\in C([0,T])$ .

The system (1.1) and (1.2) corresponds to the case $\epsilon=0$ studied in [35]. Under Assumptions 4, 6 and 7 (see Section 2), the authors used a fixed-point argument and the vanishing viscosity method to prove the existence of a solution $(u,m,\varpi)$ , where $u$ is Lipschitz and semiconcave in $x$ , and differentiable $m$ -almost everywhere, $m\in C([0,T],{\mathcal{P}}({\mathbb{R}}))$ w.r.t. the $1$ -Wasserstein distance, and $\varpi\in W^{1,1}([0,T])$ . Furthermore, under Assumption 8, they obtained uniqueness of solutions, further differentiability of $u$ in $x$ for every $x$ , and the boundedness of $u_{xx}$ and $m$ .

The connection between Hamilton-Jacobi equations and Aubtry-Mather theory is now well established; see, for example, [41], [18, 19, 20, 21], [16, 17, 5], or [45]. In particular, several generalizations of Aubry-Mather theory were developed to address problems like diffusions and study second-order Hamilton-Jacobi equations [29, 30]. In particular, duality methods, since the pioneering papers in [40] and [23] have been explored in multiple contexts, see for example [36]. Of great interest are the applications to the selection problem in the vanishing discount case, [31], [44] [34] and [33] and to the large time behavior of Hamilton-Jacobi equations [9], [37]. Recently applications of Aubry-Mather theory were developed for MFGs in [12] to study long-time behavior, and in [11], where the authors construct Mather measures to prove the existence of solutions for ergodic first-order MFG systems with state constraints.

The prototype MFG system corresponds to an optimal control problem for an agent who optimizes a cost function that depends on the aggregate behavior of other agents encoded in the population distribution $m$ . In [35], the optimal control setting of the MFG system (1.1) and (1.2) corresponds to an agent interacting with the population through the price. At the same time, the balance condition between demand and supply is satisfied. This type of interaction arises in price formation models, where the commodity price being traded is an endogenous rather than an exogenous variable.

Price formation models were studied previously in [3] and [46] in the contex of revenue maximization by a producer. Earlier price models in the context of mean-field games include [8, 7, 42, 6] and [39]. Applications to electricity markets were examined in [43, 1, 22] and [14]. Price models with a market clearing condition were introduced in [35], [2], [32], [47] and [26]. The former work addresses a model for solar renewable energy certificate markets. Finally, [27] examines the effect of a major player.

The variational problem that we consider is a relaxed version of the Lagrangian formulation introduced in [35] to derive (1.1) and (1.2). We prove a duality formula (Theorem 1.3) between solutions of the MFG system and minimizers of a variational problem in the set of generalized Mather measures. For that, we begin by introducing the Legendre transform, $L$ , of $H$ ; that is,

L(x,v)=\sup_{p\in{\mathbb{R}}}\left\{-pv-H(x,p)\right\}.

(1.3)

Our variational problem is

\inf_{\mu\in\mathcal{H}(m_{0})}\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)+vu^{\prime}_{T}(x)~{}d\mu(t,x,v),

(1.4)

where $\mathcal{H}(m_{0})$ is the set of admissible measures. These measures are Radon positive measures on $[0,T]\times{\mathbb{R}}^{2}$ that satisfy the following three conditions. First, the moment condition

\int_{[0,T]\times{\mathbb{R}}^{2}}(|x|^{\zeta_{1}}+|v|^{\zeta_{2}})d\mu(t,x,v)<\infty,

where $\zeta_{1}$ and $\zeta_{2}$ depend on the growth of the Hamiltonian in Assumption 2 and satisfy condition 3.1. Second, for some probability measure $\nu$ on ${\mathbb{R}}$ , the Radon measure verifies

\int_{[0,T]\times{\mathbb{R}}^{2}}\varphi_{t}(t,x)+v\varphi_{x}(t,x)d\mu(t,x,v)=\int_{{\mathbb{R}}}\varphi(T,x)d\nu-\int_{{\mathbb{R}}}\varphi(0,x)dm_{0}

(1.5)

for all suitable test functions $\varphi$ . We refer to the previous as the holonomy condition, as it is motivated by the holonomy condition introduced in [41]. Lastly, the admissible measures satisfy the following balance condition

\int_{[0,T]\times{\mathbb{R}}^{2}}\eta(t)(v-Q(t))d\mu(t,x,v)=0

for all $\eta$ continuous. If $u_{T}\in C^{1}({\mathbb{R}})$ with $u^{\prime}_{T}$ bounded (see Assumption 3), the holonomy condition applied to $\varphi(t,x)=u_{T}(x)$ (see (3.3)) provides the identity

\int_{[0,T]\times{\mathbb{R}}^{2}}vu^{\prime}_{T}(x)d\mu(t,x,v)=\int_{{\mathbb{R}}}u_{T}(x)d\nu-\int_{{\mathbb{R}}}u_{T}(x)dm_{0}.

Using the previous identity, the variational problem (1.4) is equivalent to

\inf_{\begin{subarray}{c}\nu\in{\mathcal{P}}({\mathbb{R}})\\ \mu\in\mathcal{H}(m_{0},\nu)\end{subarray}}\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)~{}d\mu(t,x,v)+\int_{{\mathbb{R}}}u_{T}(x)d\nu(x),

(1.6)

where $\mathcal{H}(m_{0},\nu)$ is the set of measures that satisfy the moment condition, the holonomy condition for some probability measure $\nu$ on ${\mathbb{R}}$ , and the balance condition. The difference between (1.6) and (1.4) is the term $-\int_{{\mathbb{R}}}u_{T}(x)dm_{0}$ , which is independent of $\mu$ .

The motivation for this relaxed problem is as follows. In [35], each agent selects a control variable $\alpha$ aiming to solve

\inf_{\alpha\in\mathcal{A}}\int_{0}^{T}L(x(t),\alpha(t))+\varpi(t)\alpha(t)dt+u_{T}(x(T)),

(1.7)

where $\dot{x}(t)=\alpha(t)$ , and $\mathcal{A}$ , the set of bounded measurable functions, is the set of admissible controls. The price $\varpi$ is chosen so that the aggregate supply meets the demand. Here, following Mather’s theory (see for example [31]), we introduced a relaxed version of problem (1.7). This relaxation is problem (1.6). The key idea is that each optimal trajectory $t\mapsto x^{*}(t)$ with optimal control $t\mapsto\alpha^{*}(t)$ solving (1.7) defines the measure $d\mu^{*}(t,x,v)=dt\times d_{\delta_{x^{*}(t)}}\times d_{\delta_{\alpha^{*}(t)}}$ . This measure is supported on the path $t\mapsto(x^{*}(t),\alpha^{*}(t))$ and satisfies (1.5). Accordingly, the function in (1.7) becomes

\displaystyle\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)+\varpi v~{}d\mu^{*}(t,x,v)+\int_{{\mathbb{R}}}u_{T}(x)d\nu^{*}(x)

where $d\nu^{*}=\delta_{x^{*}(T)}$ ; that is, the variational cost for the measure equals the variational cost for the optimal trajectory.

Our first result for the variational problem on measures (1.6) is a duality formula between minimizing measures and Hamilton-Jacobi equations that involves the following function. Let $h:\mathcal{P}({\mathbb{R}})\times\mathcal{P}({\mathbb{R}})\to{\mathbb{R}}\cup\{+\infty\}$ be

h(m_{0},\nu)=\begin{cases}\displaystyle\inf_{\begin{subarray}{c}\mu\in\mathcal{H}(m_{0},\nu)\end{subarray}}\int_{\Omega}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v),&\text{if }\mathcal{H}(m_{0},\nu)\neq\emptyset,\\ +\infty&\text{if }\mathcal{H}(m_{0},\nu)=\emptyset.\end{cases}

(1.8)

The main assumptions on $L$ and $u_{T}$ are stated in Section 2, after which, in Section 3, we develop a framework of Mather measures suitable for the MFG system (1.1) and (1.2). Finally, in that section, we prove the following theorem.

Theorem 1.2.

Let $h$ be given by (1.8) and let $\zeta$ satisfy (2.1). Suppose Assumptions 1-4 hold. Assume that $\nu_{T}\in\mathcal{P}({\mathbb{R}})$ is such that $\mathcal{H}(m_{0},\nu_{T})\neq\emptyset$ . Then,

	$\displaystyle h(m_{0},\nu_{T})=-\inf_{\begin{subarray}{c}\varphi,\eta\end{subarray}}\sup_{(t,x)}\bigg{(}$	$\displaystyle-\int_{\mathbb{R}}\varphi(0,x)dm_{0}(x)+\int_{\mathbb{R}}\varphi(T,x)d\nu_{T}(x)$
		$\displaystyle+T\big{(}-\varphi_{t}+Q\eta+H(x,\varphi_{x}+\eta+u^{\prime}_{T})\big{)}\bigg{)},$

where $(t,x)\in[0,T]\times{\mathbb{R}}$ , $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ and $\eta\in C([0,T])$ .

The previous result is proved in Section 3 using Fenchel-Rockafellar’s duality theorem.

Next, in Section 4, we establish additional results for the MFG system (1.1) and (1.2). In particular, in Proposition 4.2, we prove that $\varpi$ solving (1.1) and (1.2) is Lipschitz continuous. This result was stated but not proved in [35]. Here, we give the full details of the proof.

Finally, in Section 5, we establish our main result, which is summarized in the following theorem.

Theorem 1.3.

Let $(u,m,\varpi)$ solve (1.1) and (1.2). Suppose that Assumptions 1-8 hold. Then,

\int_{{\mathbb{R}}}\left(u(0,x)-u_{T}(x)\right)dm_{0}(x)-\int_{0}^{T}Q(t)\varpi(t)dt=\inf_{\begin{subarray}{c}\mu\in\mathcal{H}(m_{0})\end{subarray}}\int_{\Omega}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v).

In the previous theorem, the value of (1.4) is characterized by the solution of the MFG system (1.1) and (1.2). Although $m$ does not appear explicitly on the right-hand side of the previous expression, it determines the balance condition for the MFG. Notice that for this minimization problem, $u_{T}$ is fixed, whereas the terminal measure $\nu_{T}$ is varying (see Section 3).

2. Assumptions

Here, we present the main assumptions used in this paper. First, we consider the usual convexity assumption on the Hamiltonian, $H$ , for which we require the strongest form of this property.

Assumption 1.

For all $x\in{\mathbb{R}}$ , the map $p\mapsto H(x,p)$ is uniformly convex; that is, there exists a constant $\kappa>0$ such that $H_{pp}(x,p)\geqslant\kappa$ for all $(x,p)\in{\mathbb{R}}^{2}$ .

The previous assumption guarantees not only convexity but also coercivity of $H$ in the $p$ variable (see [4], Corollary 11.17). Hence, the Legendre transform of $H$ , given by (1.3), is well-defined, and it is convex and coercive in the second argument ([10], Theorem A.2.6).

The following four assumptions are used in Section 3 to establish duality results. The following growth conditions for $H$ and the regularity for $u_{T}$ and $Q$ are required used when we apply Fenchel-Rockafellar’s theorem.

Assumption 2.

There exists $\gamma_{1}\geqslant 1$ , $\gamma_{2}>1$ , a positive constant $C$ , and non-negative constants $C_{1}$ and $C_{2}$ such that, for all $(x,p)\in{\mathbb{R}}^{2}$ ,

\begin{cases}-C_{2}|x|^{\gamma_{1}}+\dfrac{|p|^{\gamma_{2}}}{C\gamma_{2}}-C\leqslant H(x,p)\leqslant-C_{1}|x|^{\gamma_{1}}+\dfrac{|p|^{\gamma_{2}}C}{\gamma_{2}}+C,\\ |H_{x}(x,p)|\leqslant C(|p|^{\gamma_{2}}+1),\\ |H_{p}(x,p)|\leqslant C(|p|^{\gamma_{2}-1}+1).\end{cases}

Remark 2.1.

Under Assumption 1, the Lagrangian, $L$ , defined by (1.3), satisfies (see [10], Theorem A.2.6)

v=-H_{p}(x,p)\mbox{ if and only if }p=-L_{v}(x,v).

Furthermore, Assumption 2 implies a growth condition on $L$ ; that is,

C_{1}|x|^{\gamma_{1}}+\dfrac{|v|^{\gamma_{2}^{\prime}}}{\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}}-C\leqslant L(x,v)\leqslant C_{2}|x|^{\gamma_{1}}+\dfrac{|v|^{\gamma_{2}^{\prime}}C^{\gamma_{2}^{\prime}/\gamma_{2}}}{\gamma_{2}^{\prime}}+C,

(2.1)

where $1/\gamma_{2}+1/\gamma_{2}^{\prime}=1$ . To see this, note that the first condition in Assumption 2 bounds the Legendre transform of $H$ between the one of the functions

p\mapsto-C_{1}|x|^{\gamma_{1}}+\dfrac{|p|^{\gamma_{2}}C}{\gamma_{2}}+C\quad\mbox{and}\quad p\mapsto-C_{2}|x|^{\gamma_{1}}+\dfrac{|p|^{\gamma_{2}}}{C\gamma_{2}}-C.

Their transforms are the lower and upper bounds in (2.1), respectively.

Assumption 3.

The terminal cost satisfies $u_{T}\in C^{1}({\mathbb{R}})$ , and $|u^{\prime}_{T}|\leqslant C$ for some $C>0$ .

For the supply, we assume it is a smooth function of time.

Assumption 4.

The supply function, $Q$ , is $C^{\infty}([0,T])$ .

The existence of generalized measures minimizing our variational problem (1.4) relies on the moment estimates that we impose for the initial distribution (see Proposition 5.2).

Assumption 5.

The initial density, $m_{0}$ , is a probability measure in ${\mathbb{R}}$ , and it has a finite absolute moment of order $\gamma>\gamma_{1}$ ; that is,

\int_{{\mathbb{R}}}|x|^{\gamma}m_{0}(x){\rm d}x<+\infty.

Following [35], we guarantee the solvability of (1.1) and (1.2) by considering, together with Assumption 4, the following conditions.

Assumption 6.

The Hamiltonian $H$ is separable; that is,

H(x,p)=\mathcal{H}(p)-V(x),

where $V\in C^{2}({\mathbb{R}})$ is bounded from below and $|\mathcal{H}_{pp}|,|\mathcal{H}_{ppp}|\leqslant C$ for some constant $C>0$ .

Remark 2.2.

Under the previous assumption, $L$ , defined by (1.3), is separable as well; that is

L(x,v)=\mathcal{L}(v)+V(x),

where $\mathcal{L}$ is the Legendre transform of $\mathcal{H}$ . Recalling that the Legendre transform is an involutive transformation, in case that $\mathcal{L}$ is uniformly convex, we have $\mathcal{L}_{vv}\geqslant\kappa^{\prime}$ for some $\kappa^{\prime}>0$ . Hence, ([10], Corollary A. 2.7)

\mathcal{H}_{pp}\leqslant\frac{1}{\kappa^{\prime}}.

Furthermore, under Assumption 1, we obtain $\kappa<\mathcal{H}_{pp}\leqslant 1/\kappa^{\prime}$ . By abuse of notation, we set $\mathcal{H}=H$ and $\mathcal{L}=L$ when Assumption 6 holds.

Assumption 7.

The potential $V$ , the terminal cost $u_{T}$ , the initial density function $m_{0}$ are $C^{2}({\mathbb{R}})$ functions and $V$ , $u_{T}$ are globally Lipschitz. Furthermore, there exists a constant $C>0$ such that

|V^{\prime\prime}|\leqslant C,\quad|u_{T}^{\prime\prime}|\leqslant C,\quad|m_{0}^{\prime\prime}|\leqslant C.

The following condition guarantees the uniqueness of solutions of (1.1) and (1.2).

Assumption 8.

The potential $V$ and the terminal cost $u_{T}$ are convex.

Remark 2.3.

Assume the Hamiltonian, $H$ , satisfies Assumption 6, with a potential, $V$ , satisfying Assumption 7. For Assumption 2 to hold, $V$ has to satisfy $C\geqslant\mbox{Lip}(V)$ and the growth condition

C_{1}|x|^{\gamma_{1}}-K\leqslant V(x)\leqslant C_{2}|x|^{\gamma_{1}}+K

(2.2)

for some $K>0$ , whereas $\mathcal{H}$ has to satisfy $|\mathcal{H}_{p}(p)|\leqslant C(|p|^{\gamma_{2}-1}+1)$ and the growth condition

\dfrac{|p|^{\gamma_{2}}}{C\gamma_{2}}-C\leqslant H(p)\leqslant\dfrac{|p|^{\gamma_{2}}C}{\gamma_{2}}+C.

For instance, the Hamiltonian

H(x,p)=\left(1+|p|^{2}\right)^{\frac{\gamma_{2}}{2}}-V(x)

satisfies all the assumptions above if $V$ is a globally Lipschitz function that satisfies (2.2).

3. Duality results

This section considers generalized holonomic measures for time-dependent problems in ${\mathbb{R}}$ that are compatible with the integral constraint imposed by the balance condition. We use this formulation to prove Theorem 1.2 and for the proof of Theorem 1.3 in Section 5.

Fix $T>0$ . For $\gamma_{1}\geqslant 1$ and $\gamma_{2}>1$ (see Assumption 2), let $\zeta=(\zeta_{1},\zeta_{2})$ , where

0<\zeta_{1}\leqslant\gamma_{1},\quad\mbox{and}\quad 1<\zeta_{2}<\gamma^{\prime}_{2}.

(3.1)

Let $\Omega=[0,T]\times{\mathbb{R}}\times{\mathbb{R}}$ . Let $\mathcal{R}(\Omega)$ be the set of signed Radon measures on $\Omega$ , $\mathcal{R}^{+}(\Omega)$ be the subset of non-negative elements of $\mathcal{R}(\Omega)$ ([24], page 212 and 222 or [15], Definition 1.9), and ${\mathcal{P}}({\mathbb{R}})$ be the set of probability measures on ${\mathbb{R}}$ . We define

\displaystyle\mathcal{H}_{1}=\left\{\mu\in\mathcal{R}^{+}(\Omega):~{}\int_{\Omega}(|x|^{\zeta_{1}}+|v|^{\zeta_{2}})d\mu(t,x,v)<\infty\right\}.

(3.2)

This set is determined by the growth conditions for the Hamiltonian, as in Assumption 2. Next, let

\Lambda([0,T]\times{\mathbb{R}})=\left\{\varphi\in C^{1}([0,T]\times{\mathbb{R}}):~{}\varphi_{t},~{}\varphi_{x}\in L^{\infty}([0,T]\times{\mathbb{R}})\right\}.

Notice that elements of $\Lambda([0,T]\times{\mathbb{R}})$ are globally Lipschitz continuous functions. This set corresponds to the set of test functions for the holonomy condition, which we define next. Given $m_{0},~{}\nu\in{\mathcal{P}}({\mathbb{R}})$ , let

\mathcal{H}_{2}(m_{0},\nu)=\left\{\mu\in\mathcal{R}^{+}(\Omega):~{}\begin{matrix}\int_{\Omega}\varphi_{t}(t,x)+v\varphi_{x}(t,x)d\mu(t,x,v)\\ =\int_{{\mathbb{R}}}\varphi(T,x)d\nu-\int_{{\mathbb{R}}}\varphi(0,x)dm_{0}\end{matrix}~{}\forall\varphi\in\Lambda([0,T]\times{\mathbb{R}})\right\}.

(3.3)

As mentioned in the Introduction, we refer to the condition defining the set $\mathcal{H}_{2}(m_{0},\nu)$ as the holonomy condition. For a given $\nu\in{\mathcal{P}}({\mathbb{R}})$ , the set $\mathcal{H}_{2}(m_{0},\nu)$ may be empty. Nevertheless, as we show in Remark 3.2, there are probability measures satisfying $\mathcal{H}_{2}(m_{0},\nu)\neq\emptyset$ . In case $m_{0}$ satisfies a moment hypothesis (see Assumption 5), the identity that defines the holonomy condition is well-defined even if the terms are not finite.

Corresponding to the balance condition in (1.1), we set

\displaystyle\mathcal{H}_{3}=\left\{\mu\in\mathcal{R}^{+}(\Omega):~{}\int_{\Omega}\eta(t)(v-Q(t))d\mu(t,x,v)=0,\quad\forall\eta\in C([0,T])\right\}.

(3.4)

Finally, we define

\mathcal{H}(m_{0},\nu):=\mathcal{H}_{1}\cap\mathcal{H}_{2}(m_{0},\nu)\cap\mathcal{H}_{3},\quad\mbox{and}\quad\mathcal{H}(m_{0})=\bigcup_{\nu\in{\mathcal{P}}({\mathbb{R}})}\mathcal{H}(m_{0},\nu).

Remark 3.1.

For any $\mu\in\mathcal{H}(m_{0})$ there exists a unique $\nu\in{\mathcal{P}}({\mathbb{R}})$ such that $\mu\in\mathcal{H}(m_{0},\nu)$ . To see this, let $\mu\in\mathcal{H}(m_{0},\nu)\cap\mathcal{H}(m_{0},\tilde{\nu})$ . Let $\varphi\in C^{1}_{c}({\mathbb{R}})$ . Then, $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ , and (3.3) holds for both $\nu$ and $\tilde{\nu}$ , from which we obtain

\int_{{\mathbb{R}}}\varphi(x)d\nu(x)-\int_{{\mathbb{R}}}\varphi(x)dm_{0}(x)=\int_{{\mathbb{R}}}\varphi(x)d\tilde{\nu}(x)-\int_{{\mathbb{R}}}\varphi(x)dm_{0}(x);

that is, $\int_{{\mathbb{R}}}\varphi d\nu=\int_{{\mathbb{R}}}\varphi d\tilde{\nu}$ for any $\varphi\in C^{1}_{c}({\mathbb{R}})$ . Hence, $\nu=\tilde{\nu}$ ([24], Theorem 7.2). We denote the unique measure $\nu$ such that $\mu\in\mathcal{H}(m_{0},\nu)$ as $\nu^{\mu}$ .

Remark 3.2.

If Assumptions 4 and 5 hold, $\mathcal{H}(m_{0})$ is not empty. To see this, let $\overline{X}(t)=\int_{0}^{t}Q(s)ds$ . Define $\nu\in{\mathcal{P}}({\mathbb{R}})$ by

\int_{{\mathbb{R}}}f(x)d\nu(x)=\int_{{\mathbb{R}}}f(x+\overline{X}(T))dm_{0}(x)

(3.5)

for all $f\in C_{c}({\mathbb{R}})$ , and define $\mu\in\mathcal{R}^{+}(\Omega)$ by

\int_{\Omega}\psi(t,x,v)d\mu(t,x,v)=\int_{0}^{T}\int_{{\mathbb{R}}}\psi(t,x+\overline{X}(t),Q(t))dm_{0}(x)dt

(3.6)

for all $\psi\in C_{c}(\Omega)$ . Next, we use the following cut-off function

\theta(x)=\begin{cases}1\quad&x\in(-1,1)\\ \vartheta(x)\quad&(-2,2)\setminus(-1,1)\\ 0\quad&x\in{\mathbb{R}}\setminus(-2,2),\end{cases}

where $\vartheta$ is chosen such that $\theta\in C^{1}({\mathbb{R}})$ , $0\leqslant\theta\leqslant 1$ , and $||\theta||_{C^{1}({\mathbb{R}})}\leqslant c$ . Let

h_{n}(x,v)=\theta\left(\frac{2x}{n}\right)\theta\left(\frac{2v}{n}\right)\left(|x|^{\zeta_{1}}+|v|^{\zeta_{2}}\right)\mbox{ and }g_{n}=\mathds{1}_{n}(x)\mathds{1}_{n}(v)\left(|x|^{\zeta_{1}}+|v|^{\zeta_{2}}\right),

where $\mathds{1}_{n}$ is the characteristic function of the interval $[-n,n]$ . $g_{n}$ is a sequence of measurable functions that satisfy $0\leqslant g_{n}(x,v)\leqslant|x|^{\zeta_{1}}+|v|^{\zeta_{2}}$ and $g_{\frac{n}{2}}(x,v)\to|x|^{\zeta_{1}}+|v|^{\zeta_{2}}$ pointwise for $(x,y)\in{\mathbb{R}}^{2}$ . Although the functions $g_{n}$ are not continuous, they are Borel-measurable, and hence their integral w.r.t. $\mu$ is well-defined. Note that $g_{\frac{n}{2}}(x,v)\leqslant h_{n}(x,v)\leqslant g_{n}(x,v)$ and $h_{n}\in C_{c}(\Omega)$ . Then,

	$\displaystyle\int_{\Omega}g_{\frac{n}{2}}(x,v)d\mu(t,x,v)$	$\displaystyle\leqslant\int_{\Omega}h_{n}(x,v)d\mu(t,x,v)$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}}h_{n}(x+\overline{X}(t),Q(t))dm_{0}(x)dt$
		$\displaystyle\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}g_{n}(x+\overline{X}(t),Q(t))dm_{0}(x)dt$
		$\displaystyle\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}\|x+\overline{X}(t)\|^{\zeta_{1}}+\|Q(t)\|^{\zeta_{2}}dm_{0}(x)dt$
		$\displaystyle\leqslant\int_{0}^{T}\int_{\mathbb{R}}2^{\zeta_{1}-1}\left(\|x\|^{\zeta_{1}}+\|\overline{X}(t)\|^{\zeta_{1}}\right)+\|Q(t)\|^{\zeta_{2}}dm_{0}(x)dt$
		$\displaystyle=2^{\zeta_{1}-1}\left(T\int_{\mathbb{R}}\|x\|^{\zeta_{1}}dm_{0}(x)+\\|\overline{X}\\|^{\zeta_{1}}_{L^{\zeta_{1}}([0,T])}\right)+\\|Q\\|^{\zeta_{2}}_{L^{\zeta_{2}}([0,T])}$
		$\displaystyle\leqslant 2^{\zeta_{1}-1}T\left(\int_{\mathbb{R}}\|x\|^{\zeta_{1}}dm_{0}(x)+T^{\zeta_{1}}\\|Q\\|_{\infty}^{\zeta_{1}}\right)+T\\|Q\\|^{\zeta_{2}}_{\infty}$
		$\displaystyle=C(\zeta_{1},\zeta_{2},T,m_{0},Q),$

where $C(\zeta_{1},\zeta_{2},T,m_{0},Q)$ is finite by Assumptions 4 and 5. Using the previous inequality and the Monotone Convergence Theorem, we conclude that $\mu$ satisfies (3.2). Therefore, for any $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ , we have

\begin{split}\int_{\Omega}|\varphi_{t}(t,x)|+|v||\varphi_{x}(t,x)|d\mu(t,x,v)<\infty,\\ \int_{{\mathbb{R}}}|\varphi(T,x+\overline{X}(T))|dm_{0}(x)+\int_{{\mathbb{R}}}|\varphi(0,x)|dm_{0}(x)<\infty.\end{split}

(3.7)

Denote $\overline{M}=\max\limits_{t\in[0,T]}|\overline{X}(t)|$ and let $\phi^{n}(t,x,v)=\varphi^{n}(t,x)\theta\left(\frac{v}{n}\right)$ , where $\varphi^{n}(t,x)=\varphi(t,x)\theta\left(\frac{x}{n}\right)$ . Because $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ , from the definitions of $\phi^{n}$ , $\varphi^{n}$ , and $\theta$ , we have

	$\displaystyle\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\|\varphi^{n}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi(t,x)\theta\left(\tfrac{x}{n}\right)\right\|\leqslant Cn,$		(3.8)
	$\displaystyle\max_{(t,x,v)\in\Omega}\|\phi^{n}_{t}(t,x,v)\|\leqslant\max_{(t,x)\in\Omega}\|\varphi^{n}_{t}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi_{t}(t,x)\theta\left(\tfrac{x}{n}\right)\right\|\leqslant C,$
	$\displaystyle\|\|\phi^{n}_{x}\|\|_{C(\Omega)}\leqslant\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\|\varphi^{n}_{x}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi_{x}(t,x)\theta\left(\tfrac{x}{n}\right)+\tfrac{1}{n}\varphi(t,x)\theta^{\prime}\left(\tfrac{x}{n}\right)\right\|\leqslant C.$

Relying on these estimates from Assumption 5, we have

		$\displaystyle\left\|\int_{0}^{T}\int_{n<\|x\|<2n}\int_{n<\|v\|<2n}\phi^{n}_{t}+v\phi^{n}_{x}d\mu(t,x,v)\right\|$
		$\displaystyle\leqslant\int_{0}^{T}\int_{n<\|x\|<2n}\int_{n<\|v\|<2n}C(\|v\|+1)d\mu(t,x,v)$
		$\displaystyle\leqslant\int_{\Omega}C(\|v\|+1)\left(\theta\left(\frac{2x}{n}-3\right)+\theta\left(\frac{2x}{n}+3\right)\right)\left(\theta\left(\frac{2v}{n}-3\right)+\theta\left(\frac{2v}{n}+3\right)\right)d\mu(t,x,v)$
		$\displaystyle\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}2C(\|Q(t)\|+1)\left(\theta\left(\frac{2(x+\overline{X}(t))}{n}-3\right)+\theta\left(\frac{2(x+\overline{X}(t))}{n}+3\right)\right)dm_{0}(x)$
		$\displaystyle\leqslant TC\int_{\frac{n}{2}-\overline{M}<\|x\|<\frac{5}{2}n+\overline{M}}dm_{0}(x)=o(1).$		(3.9)

Furthermore, Assumption 5 with (3.8) implies that

		$\displaystyle\left\|\left(\int_{n}^{2n-\overline{X}(T)}+\int_{-2n-\overline{X}(T)}^{-n}\right)\varphi^{n}(T,x+\overline{X}(T))dm_{0}(x)\right\|$
		$\displaystyle\leqslant\int_{n-\overline{M}<\|x\|<2n+\overline{M}}\left\|\varphi^{n}(T,x+\overline{X}(T))\right\|dm_{0}(x)\leqslant C\int_{n-\overline{M}<\|x\|<2n+\overline{M}}\|x\|dm_{0}(x)=o(1),$
		$\displaystyle\left\|\int_{n<\|x\|<2n}\varphi^{n}(0,x)dm_{0}(x)\right\|\leqslant C\int_{n<\|x\|<2n}\|x\|dm_{0}(x)=o(1),$		(3.10)

where $o(1)\to 0$ when $n\to\infty$ . Note that $\phi^{n}\in C_{c}(\Omega)$ and $\operatorname{supp}(\phi^{n})=[0,T]\times[-2n,2n]^{2}$ . Consequently, for all $n\geqslant n_{0}$ , where $n_{0}$ satisfies $\frac{\|Q\|_{\infty}}{n_{0}}\leqslant 1$ , by (3.6), we have

		$\displaystyle\int_{0}^{T}\int_{-2n}^{2n}\int_{-2n}^{2n}\phi^{n}_{t}(t,x,v)+v\phi^{n}_{x}(t,x,v)d\mu(t,x,v)$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}}\phi^{n}_{t}(t,x+\overline{X}(t),Q(t))+Q(t)\phi^{n}_{x}(t,x+\overline{X}(t),Q(t))dm_{0}(x){\rm d}t$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}}\theta\left(\frac{Q(t)}{n}\right)\left(\varphi^{n}_{t}(t,x+\overline{X}(t))+Q(t)\varphi^{n}(t,x+\overline{X}(t))\right)dm_{0}(x)dt$
		$\displaystyle=\int_{{\mathbb{R}}}\int_{0}^{T}\frac{d}{dt}\varphi^{n}(t,x+\overline{X}(t))dm_{0}(x)dt$
		$\displaystyle=\int_{-2n-\overline{X}(T)}^{2n-\overline{X}(T)}\varphi^{n}(T,x+\overline{X}(T))dm_{0}(x)-\int_{-2n}^{2n}\varphi^{n}(0,x)dm_{0}(x).$		(3.11)

On the other hand, (3.2) and (3.2), yield

	$\displaystyle\int_{0}^{T}\int_{n<\|x\|<2n}\int_{n<\|x\|<2n}\phi^{n}_{t}(t,x)+v\phi^{n}_{x}(t,x)d\mu(t,x,v)$
	$\displaystyle-\left(\int_{n}^{2n-\overline{X}(T)}+\int_{-2n-\overline{X}(T)}^{-n}\right)\varphi^{n}(T,x+\overline{X}(T))dm_{0}(x)+\int_{n<\|x\|<2n}\varphi^{n}(0,x)dm_{0}(x)=o(1).$

Therefore, (3.2) with the definitions of $\phi^{n}$ , $\varphi^{n}$ , $\theta$ implies

\begin{split}\int_{0}^{T}\int_{-n}^{n}\int_{-n}^{n}\varphi_{t}(t,x)&+v\varphi_{x}(t,x)d\mu(t,x,v)\\ &=\int_{-n}^{n}\varphi(T,x+\overline{X}(T))dm_{0}(x)-\int_{-n}^{n}\varphi(0,x)dm_{0}(x)+o(1).\end{split}

(3.12)

With similar arguments, by using (3.5), we prove that

\begin{split}\int_{-n}^{n}\varphi(T,x+\overline{X}(T))dm_{0}(x)&-\int_{-n}^{n}\varphi(0,x)dm_{0}(x)\\ &=\int_{-n}^{n}\varphi(T,x)d\nu(x)-\int_{-n}^{n}\varphi(0,x)dm_{0}(x)+o(1).\end{split}

(3.13)

Combining (3.12) and (3.13), we obtain

\begin{split}\int_{0}^{T}\int_{-n}^{n}\int_{-n}^{n}\varphi_{t}(t,x)&+v\varphi_{x}(t,x)d\mu(t,x,v)\\ &=\int_{-n}^{n}\varphi(T,x)d\nu(x)-\int_{-n}^{n}\varphi(0,x)dm_{0}(x)+o(1).\end{split}

Letting $n\to\infty$ in the preceding identity and using (3.7), we conclude that $\mu$ satisfies (3.3).

Lastly, proceeding as before, we prove that $\mu$ verifies (3.4). Hence, $\mu\in\mathcal{H}(m_{0},\nu)$ and, therefore, $\mathcal{H}(m_{0})\neq\emptyset$ .

The minimization in (1.4) is an infinite-dimensional optimization problem. To study the connection between solutions of (1.1) and the dual problem of (1.4), we compute the dual problem using Fenchel-Rockafellar’s theorem ([48], Theorem 1.9.):

Theorem 3.3.

Let $E$ be a normed vector space and let $E^{*}$ be its topological dual space. Let $f$ and $g$ be convex functions on $E$ with values in ${\mathbb{R}}\cup\{+\infty\}$ . Denote by $f^{*}$ and $g^{*}$ the Legendre-Fenchel transforms of $f$ and $g$ , respectively, defined by

f^{*}(x^{*})=\sup_{x\in E}\left(\left\langle x^{*},x\right\rangle-f(x)\right),\quad g^{*}(x^{*})=\sup_{x\in E}\left(\left\langle x^{*},x\right\rangle-g(x)\right).

Assume there exists $x_{0}\in E$ such that $f(x_{0}),~{}g(x_{0})<+\infty$ , and $f$ is continuous at $x_{0}$ . Then

\inf_{x\in E}f(x)+g(x)=\max_{y\in E^{*}}-f^{*}(-y)-g^{*}(y).

(3.14)

In the previous result, it is part of the theorem that the supremum in the right-hand side of (3.14) is a maximum.

Now, we introduce the definitions we need to apply Theorem 3.3. Recall that $\Omega=[0,T]\times{\mathbb{R}}\times{\mathbb{R}}$ , and let $\zeta=(\zeta_{1},\zeta_{2})$ according to (3.1). Consider the normed vector space

	$\displaystyle C_{\zeta}(\Omega):=$	$\displaystyle\left\{\phi\in C(\Omega):~{}\\|\phi\\|_{\zeta}:=\sup_{\Omega}\left\|\frac{\phi(t,x,v)}{1+\|x\|^{\zeta_{1}}+\|v\|^{\zeta_{2}}}\right\|<\infty,\right.$		(3.15)
		$\displaystyle\left.~{}\lim_{\|x\|,\|v\|\to\infty}\frac{\phi(t,x,v)}{1+\|x\|^{\zeta_{1}}+\|v\|^{\zeta_{2}}}=0~{}\mbox{uniformly for }t\in[0,T]\right\}.$

Remark 3.4.

Let $\zeta$ satisfy (3.1). The dual of $\left(C_{\zeta}(\Omega),\|\cdot\|_{\zeta}\right)$ is

\displaystyle\mathcal{U}^{\zeta}=\left\{\mu\in\mathcal{R}(\Omega):~{}\int_{\Omega}(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})d|\mu|(t,x,v)<\infty\right\}.

To see this, let

C_{0}(\Omega):=\left\{\psi\in C(\Omega):~{}\lim_{|x|,|v|\to\infty}\psi(t,x,v)=0,~{}\mbox{uniformly for }t\in[0,T]\right\}.

From the Riesz Representation Theorem ([24], Theorem 7.17), we have that

C_{0}(\Omega)^{*}\mbox{ and }\mathcal{R}(\Omega)\mbox{ are isomorphic}.

(3.16)

Define $\Phi:C_{0}(\Omega)\to C_{\zeta}(\Omega)$ by $\Phi(\psi)=\phi:=(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})\psi$ . Then $\Phi$ is a linear isometry since $\|\Phi(\psi)\|_{\zeta}=\|\psi\|_{\infty}$ . Now, given $f\in C_{\zeta}(\Omega)^{*}$ , define $F\in C_{0}(\Omega)^{*}$ by $F=f\circ\Phi$ . Using (3.16), there exists $\tilde{\mu}\in\mathcal{R}(\Omega)$ such that

\langle F,\psi\rangle=\int_{\Omega}\psi(t,x,v)~{}d\tilde{\mu}(t,x,v)

for all $\psi\in C_{0}(\Omega)$ . Given $\phi\in C_{\zeta}(\Omega)$ , let $\psi=\Phi^{-1}(\phi)=\frac{\phi}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}\in C_{0}(\Omega)$ . Then

\langle f,\phi\rangle=\langle F,\psi\rangle=\int_{\Omega}\phi(t,x,v)~{}\frac{d\tilde{\mu}(t,x,v)}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}.

Hence, because $(x,v)\mapsto\frac{1}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}$ is continuous and bounded, the measure $d\mu(t,x,v):=\frac{1}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}d\tilde{\mu}(t,x,v)$ is a Borel measure finite on compact sets. Therefore ([24], Theorem 7.8), $\mu$ is a Radon measure on $\Omega$ . Notice that any Hahn, and therefore, Jordan decomposition of $\tilde{\mu}$ ([24], Theorem 3.4) provides a corresponding decomposition for $\mu$ , from which we obtain that $d|\mu|=\frac{1}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}d|\tilde{\mu}|$ . Therefore,

\int_{\Omega}(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})d|\mu|(t,x,v)=\int_{\Omega}d|\tilde{\mu}|(t,x,v)<\infty.

On the other hand, any $\mu\in\mathcal{U}^{\zeta}$ defines a linear map on $C_{\zeta}(\Omega)$ by

\phi\mapsto\int_{\Omega}\phi(t,x,v)d|\mu|.

From the following inequality

\left|\int_{\Omega}(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})\dfrac{\phi(t,x,v)}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}d|\mu|\right|\leqslant\|\phi\|_{\zeta}\int_{\Omega}(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})d|\mu|,

we see that this linear map is also bounded. Hence, we conclude that $C_{\zeta}(\Omega)^{*}$ and $\mathcal{U}^{\zeta}(\Omega)$ are isomorphic. It can be proved that they are isometrically isomorphic (see [24], Theorem 7.17).

Define (see Remark 3.4)

\displaystyle\mathcal{U}_{1}=\left\{\mu\in\mathcal{U}^{\zeta}:~{}\mu\geqslant 0,~{}\int_{\Omega}d\mu=T\right\}.

(3.17)

Notice that $\mathcal{U}_{1}$ is the set of non-negative Radon measures that satisfy (3.2) and for which (3.3) holds for $\varphi(t,x)=t$ . Now, we define an operator related to the left-hand side of (3.3). Take $v\in{\mathbb{R}}$ . Define, $A^{v}:~{}C^{1}([0,T]\times{\mathbb{R}})\to C_{\zeta}(\Omega)$ by

\varphi\mapsto A^{v}\varphi=-\varphi_{t}-v\varphi_{x}.

Indeed, because $\varphi_{t},~{}\varphi_{x}\in C([0,T]\times{\mathbb{R}})$ , and

\frac{|A^{v}\varphi|}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}\leqslant\|\varphi\|_{C^{1}}\frac{1+|v|}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}\leqslant\|\varphi\|_{C^{1}}\left(1+\sup_{v\in{\mathbb{R}}}\frac{|v|}{1+|v|^{\zeta_{2}}}\right)\leqslant C\|\varphi\|_{C^{1}},

we have $A^{v}\varphi\in C_{\zeta}(\Omega)$ and $A^{v}$ is bounded. Therefore, $A^{v}$ is a linear and bounded map. We use this map to define the following sets. Let $\mathcal{C}\subset C_{\zeta}(\Omega)$ be the closed subspace

	$\displaystyle\mathcal{C}=\mbox{cl}_{\\|\cdot\\|_{\zeta}}\left\{\phi\in C_{\zeta}(\Omega):~{}\phi(t,x,v)=A^{v}\varphi(t,x)-(v-Q(t))\eta(t)~{}\mbox{ for some }\right.$		(3.18)
	$\displaystyle\left.\varphi\in\Lambda([0,T]\times{\mathbb{R}}),~{}\eta\in C([0,T])\right\},$

where $Q$ satisfies Assumption 4, and $\mbox{cl}_{\|\cdot\|_{\zeta}}$ denotes the closure with respect to $\|\cdot\|_{\zeta}$ . Notice that $\mathcal{C}$ is convex because $A^{v}$ is linear.

Given a linear and bounded operator $B:C([0,T]\times{\mathbb{R}})\to{\mathbb{R}}$ , let

\displaystyle\mathcal{U}_{2}(B)=\mbox{cl}_{\mbox{{\tiny weak}}}\left\{\mu\in\mathcal{U}^{\zeta}:~{}\int_{\Omega}A^{v}\varphi d\mu(t,x,v)=B\varphi,~{}\forall\varphi\in\Lambda([0,T]\times{\mathbb{R}})\right\},

(3.19)

where $\mbox{cl}_{\mbox{{\tiny weak}}}$ denotes the closure with respect to weak convergence of measures ([15], Definition 1.31.). The choice of the operator $B$ determines whether $\mathcal{U}_{2}(B)\neq\emptyset$ . For instance, given $\nu_{T}\in\mathcal{P}({\mathbb{R}})$ , for the operators

B\varphi=\int_{\mathbb{R}}\varphi(0,x)dm_{0}(x)-\int_{\mathbb{R}}\varphi(T,x)d\nu_{T}(x),

(3.20)

and $A^{v}$ as before, (3.19) corresponds to (3.3), and Remark 3.2 shows that $\mathcal{U}_{2}(B)\neq\emptyset$ . Analogously, (see (3.4)) we define

\displaystyle\mathcal{U}_{3}=\mbox{cl}_{\mbox{{\tiny weak}}}\left\{\mu\in\mathcal{U}^{\zeta}:~{}\int_{\Omega}\eta(t)(v-Q(t))~{}d\mu(t,x,v)=0,~{}\forall\eta\in C([0,T])\right\}.

(3.21)

Remark 3.5.

Let $B$ as in (3.20). If $m_{0},\nu\in{\mathcal{P}}({\mathbb{R}})$ are such that $\mathcal{H}(m_{0},\nu)\neq\emptyset$ , then

\mathcal{H}(m_{0},\nu)=\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}.

To see this, notice that (3.17), (3.19), and (3.21) imply $\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}\subset\mathcal{H}(m_{0},\nu)$ . For the opposite inclusion, let $\mu\in\mathcal{H}(m_{0},\nu)$ and let $A=\left\{(t,x,v)\in\Omega:~{}1<|x|^{\zeta_{1}}+|v|^{\zeta_{2}}\right\}$ . We have that $|\mu|=\mu$ because $\mu\geqslant 0$ . Writing $\int_{\Omega}d\mu=\int_{A}d\mu+\int_{A^{c}}d\mu$ , where $A^{c}$ denotes the complement of the set $A$ , we see that

\int_{A}d\mu\leqslant\int_{A}|x|^{\zeta_{1}}+|v|^{\zeta_{2}}d\mu\leqslant\int_{\Omega}|x|^{\zeta_{1}}+|v|^{\zeta_{2}}d\mu<\infty,

and $\int_{A^{c}}d\mu$ is finite because $A^{c}$ is compact and $\mu$ is a Radon measure. Hence, $\mu\in\mathcal{U}^{\zeta}$ . Moreover, since $\mu$ satisfies (3.3) and (3.4), we have that $\mu\in\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}$ . Using $\varphi(t,x)=t$ in (3.3), we obtain that $\mu\in\mathcal{U}_{1}$ . Therefore $\mu\in\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}$ .

Now, we introduce the functionals we will use in the context of the Fenchel-Rockafellar theorem. Define $f:C_{\zeta}(\Omega)\to{\mathbb{R}}\cup\{+\infty\}$ by

\displaystyle f(\phi)=T\sup_{(t,x,v)\in\Omega}\left(\phi(t,x,v)-L(x,v)-vu^{\prime}_{T}(x)\right).

(3.22)

Since $f$ is the supremum of affine functions, $f$ is convex. The following result proves continuity for this map.

Lemma 3.6.

Let $\zeta$ satisfy (3.1). Under Assumptions 1-3, the map $f$ is continuous on $C_{\zeta}(\Omega)$ .

Proof.

Let $(\phi_{n})_{n\in{\mathbb{N}}},\phi\in C_{\zeta}(\Omega)$ be such that $\lim_{n}\|\phi_{n}-\phi\|_{\zeta}=0$ . The first condition in (3.15) and the convergence of $\phi_{n}$ guarantees the existence of $C>0$ such that $\|\phi_{n}\|_{\zeta},\|\phi\|_{\zeta}\leqslant C$ for all $n$ ; that is,

|\phi_{n}(t,x,v)|,|\phi(t,x,v)|\leqslant C(1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}})\quad\mbox{for all }(t,x,v)\in\Omega,~{}n\in{\mathbb{N}}.

Let $\alpha>C$ . By Assumption 2, using (2.1) (see Remark 2.1), we have

C_{1}|x|^{\gamma_{1}}+\dfrac{|v|^{\gamma_{2}^{\prime}}}{\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}}-C\leqslant L(x,v),\quad\text{for all}\quad(x,v)\in{\mathbb{R}}^{2}.

Adding the term $vu^{\prime}_{T}(x)$ to both sides of the previous inequality, we get

\dfrac{1}{\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}}\left(\dfrac{\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}C_{1}|x|^{\gamma_{1}}+|v|^{\gamma_{2}^{\prime}}+vu_{T}^{\prime}(x)-\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}C}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}\right)\leqslant\frac{L(x,v)+vu^{\prime}_{T}(x)}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}

for all $(x,v)\in{\mathbb{R}}^{2}$ . By Assumption 3, $u^{\prime}_{T}$ is bounded. Hence, according to (3.1), the left-hand side of the previous expression goes to $+\infty$ when $|x|,|v|\to+\infty$ . Hence, we can find $r>0$ such that $|x|,|v|\geqslant r$ implies

-\frac{L(x,v)+vu^{\prime}_{T}(x)}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}\leqslant-\alpha.

Let $(x,v)\in([-r,r]^{2})^{c}$ , where $A^{c}$ denotes the complement of the set $A$ , and let $t\in[0,T]$ . Using the previous bound, we have

	$\displaystyle\phi_{n}(t,x,v)-L(x,v)-vu^{\prime}_{T}(x)$	$\displaystyle\leqslant\phi_{n}(t,x,v)-\alpha(1+\|x\|^{\zeta_{1}}+\|v\|^{\zeta_{2}})$
		$\displaystyle\leqslant(C-\alpha)(1+\|x\|^{\zeta_{1}}+\|v\|^{\zeta_{2}})$
		$\displaystyle<0,$

for $n\in{\mathbb{N}}$ . Hence,

f(\phi_{n})=T\sup_{(t,x,v)\in[0,T]\times[-r,r]^{2}}\left(\phi_{n}(t,x,v)-L(x,v)-vu^{\prime}_{T}(x)\right),

and the same holds for $\phi$ . Because the convergence on $C_{\zeta}(\Omega)$ implies uniform convergence on $[0,T]\times[-r,r]^{2}$ , we obtain

f(\phi_{n})\to f(\phi).\qed

Proposition 3.7.

Let $\zeta$ satisfy (3.1). Suppose that Assumptions 1-3 hold. Let $\mu\in\mathcal{U}^{\zeta}$ . If $\mu\ngeq 0$ then $f^{*}(\mu)=+\infty$ .

Proof.

Let $\mu\in\mathcal{U}^{\zeta}$ be such that $\mu\ngeq 0$ . Regarding $\mu$ as a linear map, by Remark 3.4, there exists $\phi\in C_{\zeta}(\Omega)$ such that $0\leqslant\phi$ and $\int_{\Omega}\phi(t,x,v)d\mu<0$ . Let $\phi_{n}=-n\phi$ , for $n\in{\mathbb{N}}$ . Thus, the sequence $(\phi_{n})_{n\in{\mathbb{N}}}$ in $C_{\zeta}(\Omega)$ satisfies

\phi_{n}\leqslant 0\quad\mbox{and}\quad\int_{\Omega}\phi_{n}d\mu\to+\infty.

(3.23)

Let $\tilde{\phi}_{n}=\phi_{n}+vu^{\prime}_{T}$ , for $n\in{\mathbb{N}}$ . By Assumption 3, we have $vu^{\prime}_{T}\in C_{\zeta}(\Omega)$ . Therefore, $\tilde{\phi}_{n}\in C_{\zeta}(\Omega)$ for $n\in{\mathbb{N}}$ . Moreover, $\int_{\Omega}\tilde{\phi}_{n}d\mu\to+\infty$ as well. From (3.22) we get

f(\tilde{\phi}_{n})=T\sup_{(t,x,v)\in\Omega}\left(\phi_{n}-L\right).

By Assumption 2, using (2.1) (see Remark 2.1) and the first condition in (3.23), we get

\phi_{n}(t,x,v)-L(x,v)\leqslant-C_{1}|x|^{\gamma_{1}}-\dfrac{|v|^{\gamma_{2}^{\prime}}}{\gamma_{2}^{\prime}C^{\gamma_{2}^{\prime}/\gamma_{2}}}+C\leqslant C.

Thus, $f(\tilde{\phi}_{n})\leqslant TC$ . Hence, we conclude that

+\infty=\lim_{n}\int_{\Omega}\tilde{\phi}_{n}d\mu-TC\leqslant\lim_{n}\int_{\Omega}\tilde{\phi}_{n}d\mu-f(\tilde{\phi}_{n})\leqslant f^{*}(\mu).\qed

Proposition 3.8.

Let $\zeta$ satisfy (3.1). Suppose that Assumptions 1-3 hold. Let $\mu\in\mathcal{U}^{\zeta}$ . If $\mu\geqslant 0$ , then

f^{*}(\mu)\geqslant\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu+\sup_{\psi\in C_{\zeta}(\Omega)}\left(\int_{\Omega}\psi d\mu-T\sup_{\Omega}\psi\right).

Proof.

From (2.1), we can add a constant $C$ to $L$ and assume that $0\leqslant L$ . Using Remark 2.1, let $L_{n}$ be a sequence in $C_{\zeta}(\Omega)$ such that $0\leqslant L_{n}\leqslant L_{n+1}\leqslant L$ and $L_{n}\to L$ pointwise. Fix $n\in{\mathbb{N}}$ , $\phi\in C_{\zeta}(\Omega)$ and let $\psi=\phi-vu^{\prime}_{T}-L_{n}\in C_{\zeta}(\Omega)$ . Then

	$\displaystyle\int_{\Omega}\phi d\mu-f(\phi)$	$\displaystyle=\int_{\Omega}\left(\psi+L_{n}+vu^{\prime}_{T}\right)d\mu-T\sup_{\Omega}\left(\psi+L_{n}-L\right)$
		$\displaystyle\geqslant\int_{\Omega}\left(\psi+L_{n}+vu^{\prime}_{T}\right)d\mu-T\sup_{\Omega}\psi.$

By the Monotone Convergence Theorem, we have $\int_{\Omega}L_{n}d\mu\to\int_{\Omega}Ld\mu$ . Therefore,

\displaystyle f^{*}(\mu)=\sup_{\phi\in C_{\zeta}(\Omega)}\int_{\Omega}\phi d\mu-f(\phi)

\displaystyle\geqslant\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu+\sup_{\psi\in C_{\zeta}(\Omega)}\left(\int_{\Omega}\psi~{}d\mu-T\sup_{\Omega}\psi\right).\qed

Proposition 3.9.

Let $\zeta$ satisfy (3.1). Suppose that Assumptions 1-3 hold. Let $f$ be as in (3.22) and let $f^{*}$ be its Legendre transform; that is,

f^{*}(\mu)=\sup_{\phi\in C_{\zeta}(\Omega)}\left(\int_{\Omega}\phi~{}d\mu-f(\phi)\right).

Then,

f^{*}(\mu)=\begin{cases}\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu&\mu\in\mathcal{U}_{1}\\ +\infty&\mbox{otherwise}\end{cases}.

Proof.

By Proposition 3.7, if $\mu\ngeq 0$ then $f^{*}(\mu)=+\infty$ . Let $\mu\geqslant 0$ . If $\mu\not\in\mathcal{U}_{1}$ , by definition, $\int_{\Omega}d\mu\neq T$ (see (3.17)). Define $\phi_{\alpha}=\alpha+vu^{\prime}_{T}-C\in C_{\zeta}(\Omega)$ , where $\alpha\in{\mathbb{R}}$ and $C$ is given by Assumption 2. Then, by (2.1), we obtain

f(\phi_{\alpha})=T\sup_{\Omega}\left(\alpha-C-L\right)\leqslant T\alpha.

Adding $\alpha\int_{\Omega}d\mu$ and rearranging the previous expression, we get

\alpha\int_{\Omega}d\mu-T\alpha\leqslant\alpha\int_{\Omega}d\mu-f(\phi_{\alpha}),

which implies that

\left(\int_{\Omega}d\mu-T\right)\sup_{\alpha\in{\mathbb{R}}}\alpha\leqslant\sup_{\alpha\in{\mathbb{R}}}\int_{\Omega}\alpha d\mu-f(\phi_{\alpha})\leqslant f^{*}(\mu).

From the preceding inequality, we conclude that $f^{*}(\mu)=+\infty$ . On the other hand, if $\mu\in\mathcal{U}_{1}$ , by definition, $\int_{\Omega}d\mu=T$ . For any $\phi\in C_{\zeta}(\Omega)$ , we have

\int_{\Omega}\phi-L-vu^{\prime}_{T}~{}d\mu\leqslant\int_{\Omega}\sup_{\Omega}\left(\phi-L-vu^{\prime}_{T}\right)~{}d\mu=T\sup_{\Omega}\left(\phi-L-vu^{\prime}_{T}\right)=f(\phi).

Rearranging the previous inequality, we obtain

\int_{\Omega}\phi d\mu-f(\phi)\leqslant\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu,

and we conclude that $f^{*}(\mu)\leqslant\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu$ . Finally, we take $\psi\equiv 0$ in Proposition 3.8 to obtain $f^{*}(\mu)\geqslant\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu$ . The result follows. ∎

Now, we define the second functional we use in the Fenchel-Rockafellar theorem. Recall the definition of $\mathcal{C}$ in (3.18). Fix (see (3.19) and (3.21))

\overline{\mu}\in\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}.

(3.24)

Define $g:C_{\zeta}(\Omega)\to{\mathbb{R}}\cup\{+\infty\}$ as

g(\phi)=\begin{cases}\displaystyle-\int_{\Omega}\phi d\overline{\mu},&\phi\in\mathcal{C}\\ +\infty,&\mbox{otherwise}.\end{cases}

(3.25)

Proposition 3.10.

Let $\zeta$ satisfy (3.1). Suppose that Assumption 4 holds. Assume that $B:C([0,T]\times{\mathbb{R}})\to{\mathbb{R}}$ is a linear and bounded operator such that $\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}\neq\emptyset$ . Then

g^{*}(\mu)=\begin{cases}0,&-\mu\in\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}\\ +\infty,&\mbox{otherwise}.\end{cases}

Proof.

Let $\mu\in\mathcal{U}^{\zeta}$ and define $\hat{\mu}=\mu+\overline{\mu}~{}\in\mathcal{R}(\Omega)$ .

Assume that $-\mu\in\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}$ . Then, $\hat{\mu}$ satisfies

\int_{\Omega}A^{v}\varphi-(v-Q(t))\eta(t)~{}d\hat{\mu}=0

for all $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ and $\eta\in C([0,T])$ . Because $\hat{\mu}$ defines a linear and bounded functional on $C_{\zeta}(\Omega)$ (see Section 3.4), the continuity under $\|\cdot\|_{\zeta}$ guarantees that $\int_{\Omega}\phi~{}d\hat{\mu}=0$ for all $\phi\in\mathcal{C}$ ; that is, $\int_{\Omega}\phi~{}d\mu=-\int_{\Omega}\phi~{}d\overline{\mu}=g(\phi)$ for all $\phi\in\mathcal{C}$ . Hence,

g^{*}(\mu)=\sup_{\phi\in C_{\zeta}(\Omega)}\left(\int_{\Omega}\phi~{}d\mu-g(\phi)\right)=\sup_{\phi\in\mathcal{C}}\left(\int_{\Omega}\phi~{}d\mu-g(\phi)\right)=0.

Now, assume that $-\mu\not\in\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}$ . Then, either $-\mu\not\in\mathcal{U}_{2}(B)$ or $-\mu\not\in\mathcal{U}_{3}$ . In the first alternative, there exists $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ such that

-\int_{\Omega}A^{v}\varphi d\mu(x,q,s)\neq B\varphi,

we have

\int_{\Omega}A^{v}\varphi d\hat{\mu}=\int_{\Omega}A^{v}\varphi d\mu+\int_{\Omega}A^{v}\varphi d\overline{\mu}\neq 0.

Define $\hat{\phi}=A^{v}\varphi$ . Then $\hat{\phi}\in\mathcal{C}$ and satisfies $\int_{\Omega}\hat{\phi}~{}d\hat{\mu}\neq 0$ , and using (3.25), we obtain

\sup_{\phi\in C_{\zeta}(\Omega)}\left(\int_{\Omega}\phi~{}d\mu-g(\phi)\right)=\sup_{\phi\in\mathcal{C}}\left(\int_{\Omega}\phi~{}d\mu-g(\phi)\right)\geqslant\int_{\Omega}\hat{\phi}~{}d\mu-g(\hat{\phi})=\int_{\Omega}\hat{\phi}~{}d\hat{\mu}.

Let $\alpha_{n}=n~{}\mbox{sgn}\left(\int_{\Omega}\hat{\phi}~{}d\hat{\mu}\right)$ , where $\mbox{sgn}(\cdot)$ denotes the sign function, and $\hat{\phi}_{n}=\alpha_{n}A^{v}\varphi$ , for $n\in{\mathbb{N}}$ . Because $\alpha_{n}\varphi$ is a sequence in $\Lambda([0,T]\times{\mathbb{R}})$ , $\hat{\phi}_{n}$ is a sequence in $\mathcal{C}$ . Furthermore, the previous inequality implies

g^{*}(\mu)\geqslant n~{}\mbox{sgn}\left(\int_{\Omega}\hat{\phi}~{}d\hat{\mu}\right)\int_{\Omega}\hat{\phi}~{}d\hat{\mu}

for all $n\in{\mathbb{N}}$ . Hence $g^{*}(\mu)=+\infty$ .

In the second alternative, there exists $\eta\in C([0,T])$ such that

\int_{\Omega}\eta(t)(v-Q(t))~{}d\mu\neq 0,

we have

\int_{\Omega}\eta(v-Q)\varphi d\hat{\mu}=\int_{\Omega}\eta(v-Q)\varphi d\mu\neq 0.

Define $\hat{\phi}=-(v-Q)\eta$ . Then $\hat{\phi}\in\mathcal{C}$ and satisfies $\int_{\Omega}\hat{\phi}~{}d\hat{\mu}\neq 0$ . Proceeding as before, we obtain $g^{*}(\mu)=+\infty$ . ∎

Theorem 3.11.

Let $\zeta$ satisfy (3.1). Suppose that Assumptions 1-4 hold. Assume that $B:C([0,T]\times{\mathbb{R}})\to{\mathbb{R}}$ is a linear and bounded operator such that $\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}\neq\emptyset$ . Then

\displaystyle\inf_{\begin{subarray}{c}\varphi,\eta\end{subarray}}\left(T\sup_{(t,x)}\bigg{(}-\varphi_{t}+Q\eta+H(x,\varphi_{x}+\eta+u^{\prime}_{T})\bigg{)}-B\varphi\right)=\max_{\mu}\left(-\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu\right),

where the supremum is taken over $(t,x)\in[0,T]\times{\mathbb{R}}$ , the infimum is taken over $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ , $\eta\in C([0,T])$ , and the maximum is taken on $\mu\in\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}$ .

Proof.

Recall that $f$ is convex, and by Lemma 3.6, $f$ is continuous on $C_{\zeta}(\Omega)$ . By definition, $g$ is convex. Therefore, to use Theorem 3.3, we need to find $\phi\in C_{\zeta}(\Omega)$ such that $f(\phi),g(\phi)<+\infty$ . Take $\varphi(t,x)=Ct-u_{T}(x)$ , where $C$ is given by Assumption 2. Then $\phi=A^{v}\varphi=-C+vu^{\prime}_{T}\in\mathcal{C}$ . By Assumption 2 and (2.1), we have

f(\phi)\leqslant 0.

From the definition of $g$ (see (3.25)),

g(\phi)=-B\varphi,

and by Assumption 3, $B\varphi$ is finite. Hence, relying on the duality relation between $C_{\zeta}(\Omega)$ and $\mathcal{U}^{\zeta}$ (see Remark 3.4), we apply Theorem 3.3 to get

\inf_{\phi\in C_{\zeta}(\Omega)}\left(f(\phi)+g(\phi)\right)=\max_{\mu\in\mathcal{U}^{\zeta}}\left(-f^{*}(-\mu)-g^{*}(\mu)\right)=\max_{\mu\in\mathcal{U}^{\zeta}}\left(-f^{*}(\mu)-g^{*}(-\mu)\right).

From Proposition 3.9 and Proposition 3.10, it follows that

\max_{\mu\in\mathcal{U}^{\zeta}}\left(-f^{*}(\mu)-g^{*}(-\mu)\right)=\max_{\mu\in\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}}\left(-\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu\right).

By (3.25),

\inf_{\phi\in C_{\zeta}(\Omega)}\left(f(\phi)+g(\phi)\right)=\inf_{\phi\in\mathcal{C}}\left(T\sup_{\Omega}\left(\phi-L-vu^{\prime}_{T}\right)-\int_{\Omega}\phi d\overline{\mu}\right),

and using the definition of $\mathcal{C}$ in (3.18), the selection of $\overline{\mu}$ in (3.24), and the definition of the Legendre transform (1.3), we obtain

		$\displaystyle\inf_{\phi\in\mathcal{C}}\left(T\sup_{\Omega}\left(\phi-L-vu^{\prime}_{T}\right)-\int_{\Omega}\phi d\overline{\mu}\right)$
	$\displaystyle=$	$\displaystyle\inf_{\begin{subarray}{c}\varphi\in\Lambda([0,T]\times{\mathbb{R}})\\ \eta\in C([0,T])\end{subarray}}\left(T\sup_{\Omega}\left(A^{v}\varphi-(v-Q)\eta-L-vu^{\prime}_{T}\right)-\int_{\Omega}A^{v}\varphi-(v-Q)\eta~{}d\overline{\mu}\right)$
	$\displaystyle=$	$\displaystyle\inf_{\begin{subarray}{c}\varphi\in\Lambda([0,T]\times{\mathbb{R}})\\ \eta\in C([0,T])\end{subarray}}\left(T\sup_{\Omega}\left(-\varphi_{t}+Q\eta-v(\varphi_{x}+\eta+u^{\prime}_{T})-L\right)-B\varphi\right)$
	$\displaystyle=$	$\displaystyle\inf_{\begin{subarray}{c}\varphi\in\Lambda([0,T]\times{\mathbb{R}})\\ \eta\in C([0,T])\end{subarray}}\left(T\sup_{(t,x)\in[0,T]\times{\mathbb{R}}}\left(-\varphi_{t}+Q\eta+\sup_{v\in{\mathbb{R}}}\left(-v(\varphi_{x}+\eta+u^{\prime}_{T})-L\right)\right)-B\varphi\right)$
	$\displaystyle=$	$\displaystyle\inf_{\begin{subarray}{c}\varphi\in\Lambda([0,T]\times{\mathbb{R}})\\ \eta\in C([0,T])\end{subarray}}\left(T\sup_{(t,x)\in[0,T]\times{\mathbb{R}}}\bigg{(}-\varphi_{t}+Q\eta+H(x,\varphi_{x}+\eta+u^{\prime}_{T})\bigg{)}-B\varphi\right).$

The result follows. ∎

Now, we use the duality result from Theorem 3.11 to prove Theorem 1.2.

Proof of Theorem 1.2.

Let $B$ be given by (3.20) and consider the set $\mathcal{U}_{2}(B)$ according to (3.19). By assumption, $\mathcal{H}(m_{0},\nu_{T})\neq\emptyset$ , from which (1.8) and Remark 3.5 imply

h(m_{0},\nu_{T})=\inf_{\begin{subarray}{c}\mu\in\mathcal{H}(m_{0},\nu_{T})\end{subarray}}\int_{\Omega}L+vu_{T}^{\prime}d\mu=\inf_{\mu\in\mathcal{U}_{1}\cap\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}}\int_{\Omega}L+vu^{\prime}_{T}~{}d\mu.

In particular, $\mathcal{U}_{2}(B)\cap\mathcal{U}_{3}\neq\emptyset$ . The conclusion follows by invoking Theorem 3.11 and the previous equality. ∎

4. Preliminary results on MFG

Here, we consider approximations of Lipschitz continuous solutions of the Hamilton-Jacobi equation in (1.1). We provide a commutation lemma, which states that the approximated solutions are sub-solutions of an approximate Hamilton-Jacobi equation. Then, we improve the result in [35], where the authors proved that $\varpi$ solving (1.1) and (1.2) satisfies $\varpi\in W^{1,1}([0,T])$ . A better result can be established as $\varpi$ is Lipschitz continuous, as we prove here. This result, in turn, enables the use of the commutation lemma.

4.1. A commutation lemma

The commutation lemmas presented in [37] and [45] are applied to a Hamilton-Jacobi equation where the state variable is constrained to the $d$ -dimensional torus; that is, periodic boundary conditions. Here, we present a version of this lemma that is valid for the non-periodic case and takes into account the dependence of the Hamilton-Jacobi equation on the price variable.

We start by introducing smooth approximations to the solutions of (1.1). Let $\rho,\theta\in C^{\infty}_{c}({\mathbb{R}};[0,\infty))$ be symmetric standard mollifiers, i.e.

\operatorname{supp}\rho,\operatorname{supp}\theta\subset[-1,1],~{}\rho(t)=\rho(-t),~{}\theta(x)=\theta(-x),~{}\mbox{and}~{}\|\rho\|_{L^{1}({\mathbb{R}})}=\|\theta\|_{L^{1}({\mathbb{R}})}=1.

For $0<\alpha<T$ , set $\rho^{\alpha}(t):=\alpha^{-1}\rho(\alpha^{-1}t),\,t\in{\mathbb{R}}$ and $\theta^{\alpha}(x):=\alpha^{-1}\theta(\alpha^{-1}x),\,x\in{\mathbb{R}}$ . Then, we have that $\|\rho^{\alpha}\|_{L^{1}({\mathbb{R}})}=\|\theta^{\alpha}\|_{L^{1}({\mathbb{R}})}=1$ , and

\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)|y|~{}dyds,~{}\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)s~{}dyds\leqslant\alpha.

(4.1)

For $w\in C([\alpha,T]\times{\mathbb{R}})$ , define $w^{\alpha}\in C^{\infty}([\alpha,T]\times{\mathbb{R}})$ as

w^{\alpha}(t,x)=\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)w(t-s,x-y)dyds,\quad(t,x)\in[\alpha,T]\times{\mathbb{R}}.

(4.2)

Lemma 4.1.

Suppose that Assumptions 1 and 2 hold. Let $(w,m,\varpi)$ solve (1.1) (see Remark 1.1). Assume further that $w$ is Lipschitz in $x$ and $\varpi$ is Lipschitz. Let $w^{\alpha}$ be defined as in (4.2). Then, there exists $C^{\prime}>0$ depending on $\varpi$ , $H$ and the Lipschitz constants of $w$ and $\varpi$ such that

-w^{\alpha}_{t}+H(x,\varpi+w^{\alpha}_{x})\leqslant C^{\prime}\alpha,\quad\mbox{for all}\quad(t,x)\in[\alpha,T]\times{\mathbb{R}}.

(4.3)

Proof.

To obtain the desired inequality, we write the left-hand side of (4.3) as a convolution between $\rho^{\alpha}\theta^{\alpha}$ and the left-hand side of the first equation in (1.1). Thus, for the first term, we have

-w^{\alpha}_{t}(t,x)=\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)(-w_{t}(t-s,x-y))dyds.

(4.4)

For the second term, by Jensen’s inequality ([28], Theorem 204), we have

	$\displaystyle H(x,\varpi(t)+w^{\alpha}_{x}(t,x))$	$\displaystyle=H\left(x,\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)(\varpi(t)+w_{x}(t-s,x-y))dyds\right)$
		$\displaystyle\leqslant\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)H(x,\varpi(t)+w_{x}(t-s,x-y))dyds.$		(4.5)

Let $t\in[\alpha,T]$ , $s\in[0,\alpha]$ , $x,y\in{\mathbb{R}}$ , and

q(t,x;s,y):=H\big{(}x,\varpi(t)+w_{x}(t-s,x-y)\big{)}-H\big{(}x-y,\varpi(t-s)+w_{x}(t-s,x-y)\big{)}.

Using Assumption 2 and the Lipschitz continuity of $w$ and $\varpi$ , we get

	$\displaystyle\|q(t,x;s,y)\|$	$\displaystyle\leqslant\left\|H\big{(}x,\varpi(t)+w_{x}(t-s,x-y))-H(x,\varpi(t-s)+w_{x}(t-s,x-y)\big{)}\right\|$
		$\displaystyle+\left\|H\big{(}x,\varpi(t-s)+w_{x}(t-s,x-y))-H(x-y,\varpi(t-s)+w_{x}(t-s,x-y)\big{)}\right\|$
		$\displaystyle\leqslant C\|\varpi(t)-\varpi(t-s)\|\left(\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}-1}+1\right)$
		$\displaystyle+C\|y\|\big{(}\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}}+1\big{)}$
		$\displaystyle\leqslant C^{\prime}s\left(\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}-1}+1\right)$
		$\displaystyle+C\|y\|\big{(}\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}}+1\big{)}$
		$\displaystyle\leqslant C^{\prime}(s+\|y\|),$

where $C^{\prime}$ depends on $\varpi$ , $\gamma_{2}$ , and the Lipschitz constants of $w$ and $\varpi$ . From the previous inequality and (4.1), we obtain

\displaystyle\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)q(t,x;s,y)dyds

\displaystyle\leqslant\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)C^{\prime}(s+|y|)dyds\leqslant C^{\prime}\alpha.

(4.6)

Then, from (4.1) and (4.6), we have

	$\displaystyle H(x,\varpi(t)+w^{\alpha}_{x}(t,x))$
	$\displaystyle\leqslant\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)H(x-y,\varpi(t-s)+w_{x}(t-s,x-y))dyds+C^{\prime}\alpha.$

Using the preceding inequality and (4.4), we get

	$\displaystyle-w^{\alpha}_{t}+H(x,\varpi+w^{\alpha}_{x})$
	$\displaystyle\leqslant\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)\big{(}-w_{t}(t-s,x-y)+H(x-y,\varpi(t-s)+w_{x}(t-s,x-y))\big{)}dyds$
	$\displaystyle\quad+C^{\prime}\alpha,$

which implies (4.3). ∎

4.2. Lipschitz continuity of the price

We begin by recalling the following techniques and results from [35] if Assumptions 4, 6, and 7 hold. Firstly, to prove the existence of a solution $(u,m,\varpi)$ of (1.1) and (1.2), the authors used the vanishing viscosity method, which relies on the following regularized version of (1.1)

\begin{cases}-u_{t}(t,x)+H(x,\varpi(t)+u_{x}(t,x))=\epsilon u_{xx}\\ m_{t}(t,x)-\big{(}H_{p}(x,\varpi(t)+u_{x}(t,x))m(t,x)\big{)}_{x}=\epsilon m_{xx}(t,x)\\ -\int_{{\mathbb{R}}}H_{p}(x,\varpi(t)+u_{x}(t,x))m(t,x)dx=Q(t)\end{cases}\quad(t,x)\in[0,T]\times{\mathbb{R}},

(4.7)

subject to (1.2), where $\epsilon>0$ . Secondly, the proof of existence of a solution $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ of (4.7) and (1.2) uses a fixed-point argument. This argument shows that $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ satisfies

\dot{\varpi^{\epsilon}}=\frac{-\dot{Q}-\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})H_{x}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}+\epsilon H_{ppp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})(u^{\epsilon}_{xx})^{2}m^{\epsilon}~{}dx}{\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}~{}dx},

(4.8)

and $\varpi^{\epsilon}(0)$ is determined by

\int_{{\mathbb{R}}}H_{p}(x,\varpi^{\epsilon}(0)+u^{\epsilon}(0,x))m_{0}(x)~{}dx=-Q(0).

Using (4.8), we can deduce the Lipschitz continuity of $\varpi$ , where $(u,m,\varpi)$ solves (1.1) and (1.2), as we show next.

Proposition 4.2.

Suppose that Assumptions 1, 4, 6 and 7 hold. Then, there exists a solution $(u,m,\varpi)$ of (1.1) and (1.2) such that $\varpi$ is Lipschitz continuous.

Proof.

The existence of a solution $(u,m,\varpi)$ of (1.1) and (1.2) is guaranteed by Theorem 1 in [35]. We aim to prove that $\varpi$ , obtained in [35], is Lipschitz. To obtain this solution, the authors considered, for $\epsilon>0$ , solutions $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ of (4.7) and (1.2) that satisfy (4.8). Extracting a sub-sequence if necessary, it is guaranteed that $\varpi^{\epsilon}\to\varpi$ uniformly. To prove that $\varpi$ is Lipschitz, we consider the right-hand side of (4.8). By Assumption 1, we have

\displaystyle t\mapsto\frac{1}{\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}~{}dx}\leqslant\frac{1}{\kappa}\quad\mbox{for all}\quad t\in[0,T].

(4.9)

By Assumptions 6 and 7, $|H_{x}|=|V^{\prime}|\leqslant\mbox{Lip}(V)$ , where $\mbox{Lip}(V)$ denotes the Lipschitz constant of $V$ . Hence, Assumption 6 implies that

\displaystyle t\mapsto\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})H_{x}(s,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}~{}dx\leqslant\tfrac{\mbox{Lip}(V)}{\kappa^{\prime}}\quad\mbox{for all}\quad t\in[0,T].

(4.10)

By Assumption 6 and Assumption 1, we have

	$\displaystyle\int_{0}^{T}\int_{{\mathbb{R}}}\|H_{ppp}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})\|(u_{xx}^{\epsilon})^{2}m^{\epsilon}dxdt$	$\displaystyle\leqslant C\int_{0}^{T}\int_{{\mathbb{R}}}(u_{xx}^{\epsilon})^{2}m^{\epsilon}dxdt$
		$\displaystyle\leqslant\frac{C}{\kappa}\int_{0}^{T}\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})(u_{xx}^{\epsilon})^{2}m^{\epsilon}dxdt.$		(4.11)

Assumptions 6, 7 and Proposition 5 in [35] guarantee that the term

\int_{0}^{T}\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})(u_{xx}^{\epsilon})^{2}m^{\epsilon}{\rm d}x{\rm d}t

(4.12)

has an upper bound that is independent of $\epsilon$ . Hence, using Assumption 4, (4.9), (4.10) and (4.2), we can write (4.8) as

\dot{\varpi^{\epsilon}}=\vartheta^{\epsilon}_{\infty}+\epsilon\vartheta_{1}^{\epsilon},

where

\displaystyle\vartheta_{\infty}^{\epsilon}=\frac{-\dot{Q}-\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})H_{x}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}dx}{\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}~{}dx}\in L^{\infty}([0,T]),

\displaystyle\vartheta_{1}^{\epsilon}=\frac{-\int_{{\mathbb{R}}}H_{ppp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})(u^{\epsilon}_{xx})^{2}m^{\epsilon}~{}dx}{\int_{{\mathbb{R}}}H_{pp}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}~{}dx}\in L^{1}([0,T]),

and they satisfy

\|\vartheta_{1}^{\epsilon}\|_{L^{1}([0,T])}\leqslant C^{\prime}\quad\mbox{and}\quad\|\vartheta_{\infty}^{\epsilon}\|_{L^{\infty}([0,T])}\leqslant C^{\prime}

for $\epsilon\to 0$ , where $C^{\prime}$ is independent of $\epsilon$ . Hence, ([25], Proposition 1.202) passing to a sub-sequence, there exists $\mu\in\mathcal{R}([0,T])$ such that $\vartheta^{\epsilon}_{1}$ converges in the weak- $\star$ topology to $\mu$ ; that is,

\int_{0}^{T}\vartheta^{\epsilon}_{1}\eta~{}dt\to\int_{0}^{T}\eta~{}d\mu\quad\mbox{for all}\quad\eta\in C([0,T]).

(4.13)

Passing to a further sub-sequence if necessary, ([25], Proposition 2.46) there exists $\vartheta_{\infty}\in L^{\infty}([0,T])$ such that $\vartheta_{\infty}^{\epsilon}$ converges in the weak- $\star$ topology to $\vartheta_{\infty}$ ; that is,

\int_{0}^{T}\vartheta^{\epsilon}_{\infty}\eta~{}dt\to\int_{0}^{T}\vartheta_{\infty}\eta~{}dt\quad\mbox{for all}\quad\eta\in L^{1}([0,T]).

(4.14)

Let $\eta\in C^{1}_{c}((0,T))$ . By uniform convergence, we have that

\int_{0}^{T}\left(\vartheta^{\epsilon}_{\infty}+\epsilon\vartheta_{1}^{\epsilon}\right)\eta~{}dt=\int_{0}^{T}\dot{\varpi^{\epsilon}}\eta~{}dt=-\int_{0}^{T}\varpi^{\epsilon}\dot{\eta}~{}dt\to-\int_{0}^{T}\varpi\dot{\eta}~{}dt,

and by (4.13) and (4.14), we have that

\int_{0}^{T}\left(\vartheta^{\epsilon}_{\infty}+\epsilon\vartheta_{1}^{\epsilon}\right)\eta~{}dt\to\int_{0}^{T}\vartheta_{\infty}\eta~{}dt.

Hence, $\dot{\varpi}=\vartheta_{\infty}$ in the sense of distributions. Thus, $\varpi\in W^{1,\infty}([0,T])$ , which is equivalent to ([15], Theorem 4.5) $\varpi$ being Lipschitz continuous in $[0,T]$ . ∎

5. Proof of Theorem 1.3

Here, we use the results from Sections 3 and 4 to prove Theorem 1.3. We divide the proof into two lemmas, Lemma 5.1 and Lemma 5.6.

Lemma 5.1.

Let $m_{0}\in{\mathcal{P}}({\mathbb{R}})$ . Suppose that Assumptions 1-8 hold. Let $(u,m,\varpi)$ solve (1.1) and (1.2). Then,

\int_{{\mathbb{R}}}\left(u(0,x)-u_{T}(x)\right)dm_{0}(x)-\int_{0}^{T}\varpi(t)Q(t)dt\leqslant\inf_{\begin{subarray}{c}\mu\in\mathcal{H}(m_{0})\end{subarray}}\int_{\Omega}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v).

Proof.

By Assumptions 4, 6, 7, and 8, Theorem 1 in [35] guarantees the existence of a unique $(u,m,\varpi)$ solving (1.1) and (1.2). Because $u$ is continuous (see Remark 1.1), let $w^{\alpha}$ be the function given by (4.2); that is,

w^{\alpha}(t,x)=\int_{0}^{\infty}\rho^{\alpha}(s)\int_{{\mathbb{R}}}\theta^{\alpha}(y)u(t-s,x-y)dyds,\quad(t,x)\in[\alpha,T]\times{\mathbb{R}}.

(5.1)

For $(t,x)\in[0,T]\times{\mathbb{R}}$ , set

u^{\alpha}(t,x)=w^{\alpha}\left(\tfrac{T-\alpha}{T}t+\alpha,x\right)-u_{T}(x),

which is $C^{1}([0,T]\times{\mathbb{R}})$ due to Assumption 3 and (5.1). By Assumptions 6 and 7, the map $x\mapsto u(t,x)$ is Lipschitz for $0\leqslant t\leqslant T$ ([35], Proposition 1), and the Lipschitz constant depends on $T$ and the estimates for $V$ and $u_{T}$ . Hence, $u_{x}$ is bounded independently of $t$ . Therefore, $u_{x}^{\alpha}\in L^{\infty}([0,T]\times{\mathbb{R}})$ because

	$\displaystyle u_{x}^{\alpha}(t,x)$	$\displaystyle=w^{\alpha}_{x}\left(\tfrac{T-\alpha}{T}t+\alpha,x\right)-u_{T}^{\prime}(x)$
		$\displaystyle=\int_{0}^{\infty}\rho^{\alpha}(s)\int_{\mathbb{R}}\theta^{\alpha}(y)u_{x}\left(\tfrac{T-\alpha}{T}t+\alpha-s,x-y\right)dyds-u_{T}^{\prime}(x).$

Furthermore, recalling that $u$ is a viscosity solution to the first equation in (4.7) with $\varepsilon=0$ , we have that the first equation in (4.7) with $\varepsilon=0$ holds a.e. $(t,x)\in(0,T)\times{\mathbb{R}}$ . Using this and the facts that $u_{x}\in L^{\infty}((0,T)\times{\mathbb{R}})$ and $\varpi\in W^{1,\infty}([0,T])$ , we deduce that $u_{t}\in L_{loc}^{\infty}((0,T)\times{\mathbb{R}})$ . Thus, $u_{t}^{\alpha}\in L^{\infty}((0,T)\times{\mathbb{R}})$ because

	$\displaystyle u_{t}^{\alpha}(t,x)$	$\displaystyle=\tfrac{T-\alpha}{T}w^{\alpha}_{t}\left(\tfrac{T-\alpha}{T}t+\alpha,x\right)$
		$\displaystyle=\tfrac{T-\alpha}{T}\int_{0}^{\infty}\rho^{\alpha}(s)\int_{\mathbb{R}}\theta^{\alpha}(y)u_{t}\left(\tfrac{T-\alpha}{T}t+\alpha-s,x-y\right)dyds.$

Hence, $u^{\alpha}\in\Lambda([0,T]\times{\mathbb{R}})$ . Now, take $\mu\in\mathcal{H}(m_{0})$ (see Remark 3.2). By (3.3), we have

\int_{\Omega}u^{\alpha}_{t}(t,x)+vu^{\alpha}_{x}(t,x)d\mu(t,x,v)=\int_{{\mathbb{R}}}u^{\alpha}(T,x)d\nu^{\mu}(x)-\int_{{\mathbb{R}}}u^{\alpha}(0,x)dm_{0}(x).

(5.2)

By Assumption 1 and (1.3), using $p=\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}(t,x)+u_{T}^{\prime}(x)$ , it follows that

	$\displaystyle-vu^{\alpha}_{x}(t,x)$	$\displaystyle\leqslant L(x,v)+vu_{T}^{\prime}(x)+v\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)$
		$\displaystyle\quad+H\big{(}x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}(t,x)+u_{T}^{\prime}(x)\big{)}.$		(5.3)

Moreover, by the Lipschitz continuity of $u$ in $x$ and Proposition 4.2, we apply Lemma 4.1 to $w$ defined by (5.1) to get

-w^{\alpha}_{t}\left(\tfrac{T-\alpha}{T}t+\alpha,x\right)+H\left(x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+w^{\alpha}_{x}\left(\tfrac{T-\alpha}{T}t+\alpha,x\right)\right)\leqslant C^{\prime}\alpha,

for $(t,x)\in[0,T]\times{\mathbb{R}}$ ; that is,

-\tfrac{T}{T-\alpha}u^{\alpha}_{t}+H\left(x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}\left(t,x\right)+u_{T}^{\prime}(x)\right)\leqslant C^{\prime}\alpha,\quad\text{ for }(t,x)\in[0,T]\times{\mathbb{R}}.

(5.4)

Therefore, by (5.2), (5), (5.4), and using $\varphi=t$ in (3.2), we have

	$\displaystyle\int_{{\mathbb{R}}}u^{\alpha}(0,x)dm_{0}(x)-\int_{{\mathbb{R}}}u^{\alpha}(T,x)d\nu^{\mu}(x)$
	$\displaystyle=\int_{\Omega}-u^{\alpha}_{t}(t,x)-vu^{\alpha}_{x}(t,x)d\mu(t,x,v)$
	$\displaystyle\leqslant\int_{\Omega}-u^{\alpha}_{t}d\mu(t,x,v)+\int_{\Omega}L(x,v)+vu_{T}^{\prime}d\mu(t,x,v)$
	$\displaystyle\quad+\int_{\Omega}v\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+H\left(x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}+u_{T}^{\prime}\right)d\mu(t,x,v)$
	$\displaystyle\leqslant\int_{\Omega}L(x,v)+vu_{T}^{\prime}d\mu(t,x,v)+\int_{\Omega}v\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)d\mu(t,x,v)+(T-\alpha)C^{\prime}\alpha$
	$\displaystyle\quad+\int_{\Omega}H\left(x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}+u_{T}^{\prime}\right)-\tfrac{T-\alpha}{T}H\left(x,\varpi\left(\tfrac{T-\alpha}{T}t+\alpha\right)+u^{\alpha}_{x}+u_{T}^{\prime}\right)d\mu(t,x,v).$

Taking $\alpha\to 0$ in the previous inequality and using (3.4) with $\eta=\varpi$ , we obtain

\int_{{\mathbb{R}}}\left(u(0,x)-u_{T}(x)\right)dm_{0}(x)\leqslant\int_{\Omega}L(x,v)+vu_{T}^{\prime}d\mu(t,x,v)+\int_{\Omega}Q(t)\varpi(t)d\mu(t,x,v).

(5.5)

Finally, taking $\varphi(t,x)=\int_{0}^{t}Q(s)\varpi(s)ds$ in (3.3), we have

\int_{\Omega}Q(t)\varpi(t)d\mu(t,x,v)=\int_{0}^{T}Q(t)\varpi(t)dt.

Hence, (5.5) becomes

\int_{{\mathbb{R}}}\left(u(0,x)-u_{T}(x)\right)dm_{0}(x)-\int_{0}^{T}Q(t)\varpi(t)dt\leqslant\int_{\Omega}L(x,v)+vu_{T}^{\prime}d\mu(t,x,v).

Since $\mu\in\mathcal{H}(m_{0})$ is arbitrary, the preceding inequality completes the proof. ∎

For the second part of the proof of Theorem 1.3, we rely on (4.7), the regularized version of (1.1), subject to (1.2). We recall that, if Assumptions 4, 6 and 7 hold, ([35], Theorem 1) there exists a solution $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ of (4.7) and (1.2), where $u^{\epsilon}$ is a viscosity solution of the first equation, Lipschitz and semiconcave in $x$ , and differentiable $m^{\epsilon}$ -almost everywhere, $m^{\epsilon}\in C([0,T],{\mathcal{P}}({\mathbb{R}}))$ w.r.t. the $1$ -Wasserstein distance, and $\varpi^{\epsilon}\in W^{1,1}([0,T])$ is continuous. Moreover, if $\epsilon>0$ or $\epsilon=0$ and Assumption 8 holds, this solution is unique. Using the previous results for the solution of (4.7) when $\epsilon>0$ , we take $\epsilon\to 0$ to exhibit a measure $\mu\in\mathcal{H}(m_{0})$ for which the inequality

\int_{{\mathbb{R}}}u(0,x)-u_{T}(x)dm_{0}(x)-\int_{0}^{T}\varpi(t)Q(t)dt\geqslant\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v)

holds. We begin by establishing the following moment estimate for the probability measures $m^{\epsilon}$ when $\epsilon>0$ .

Proposition 5.2.

Suppose Assumptions 1, 2, 4, 6-8 hold. Assume further that $m_{0}$ satisfies Assumption 5. Then, for $0<\epsilon<1$ , the solution $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ of (4.7)-(1.2) satisfies

\int_{{\mathbb{R}}}|x|^{\gamma}m^{\epsilon}(t,x)dx<C\quad\mbox{for almost every}~{}t\in[0,T],

where the constant $C$ is independent of $\epsilon$ .

Proof.

By Assumptions 6, 7 and 8, there exist a unique solution $(u^{\epsilon},m^{\epsilon},\varpi^{\epsilon})$ of (4.7) and (1.2) ([35], Theorem 1). Then, by Assumptions 1, 2, 4, 6 and 7, and the bounds on $\varpi^{\epsilon}$ and $u_{x}^{\epsilon}$ (see Proposition 4.2 and [35], Propositions 1 and 6), we have, for $0<\epsilon<1$ ,

$\displaystyle\|H_{p}(\varpi^{\epsilon}(t)+u^{\epsilon}_{x}(t,x))\|$	$\displaystyle\leqslant C\left(\|\varpi^{\epsilon}(t)+u^{\epsilon}_{x}(t,x)\|^{\gamma_{2}-1}+1\right)$
	$\displaystyle\leqslant C\left(C^{\prime}(\gamma_{2})\left(\\|\varpi^{\epsilon}\\|_{\infty}^{\gamma_{2}-1}+\mbox{Lip}(u^{\epsilon})^{\gamma_{2}-1}\right)+1\right)$
	$\displaystyle\leqslant C\left(C^{\prime}(\gamma_{2})\left(\epsilon^{\gamma_{2}-1}C^{\prime}+\mbox{Lip}(u^{\epsilon})^{\gamma_{2}-1}\right)+1\right)$
	$\displaystyle=C_{1}\epsilon^{\gamma_{2}-1}+C_{2}$
	$\displaystyle\leqslant\tilde{C},$	(5.6)

where $C^{\prime}(\gamma_{2})=\max\{2^{\gamma_{2}-2},1\}$ and $\mbox{Lip}(u^{\epsilon})$ , and therefore $C_{1}$ , $C_{2}$ , and $\tilde{C}$ are independent of $\varpi^{\epsilon}$ and $\epsilon$ . Furthermore, $u^{\epsilon}$ defines the optimal feedback in a stochastic optimal control problem, for which the optimal trajectory satisfies

\mathrm{d}\mathrm{x}_{t}=-H_{p}(\mathrm{x}_{t},\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,\mathrm{x}_{t}))\mathrm{d}t+\sqrt{2\epsilon}\mathrm{d}W_{t},

(5.7)

where $W_{t}$ is a one-dimensional Brownian motion (see [35]). Using Assumptions 2 and 6, the vector field

(t,x)\mapsto H_{p}(x,\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,x))=H_{p}(\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,x))

is bounded and uniformly Lipschitz. Hence, $m(t,\cdot)=\mathcal{L}(\mathrm{x}_{t})$ , where $\mathcal{L}(\mathrm{x})$ denotes the law of the random variable $\mathrm{x}$ , is a weak solution of the second equation in (4.7) ([13], Lemma 4.2.3), and by Assumption 8, this weak solution is unique. Hence $m^{\epsilon}(t,\cdot)=\mathcal{L}(\mathrm{x}_{t})$ . Writing (5.7) as

\mathrm{x}_{t}=x+\int_{0}^{t}-H_{p}(\mathrm{x}_{t},\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,\mathrm{x}_{t}))\mathrm{d}t+\int_{0}^{t}\sqrt{2\epsilon}\mathrm{d}W_{t},

where $x\in{\mathbb{R}}$ , and using (5), we have, for $0<\epsilon<1$ ,

	$\displaystyle\|\mathrm{x}_{t}\|^{\gamma}$	$\displaystyle\leqslant 2^{{}^{\gamma}-1}\left(\|x\|^{\gamma}+2^{\gamma-1}\left(T^{\gamma}\tilde{C}^{\gamma}+\sqrt{2\epsilon}^{\gamma}\|W_{t}\|^{\gamma}\right)\right)$
		$\displaystyle\leqslant 2^{{}^{\gamma}-1}\|x\|^{\gamma}+C_{1}^{\prime}+C_{2}^{\prime}\|W_{t}\|^{\gamma},$		(5.8)

where $C_{1}^{\prime}$ and $C_{2}^{\prime}$ are independent of $\varpi^{\epsilon}$ and $\epsilon$ . Because $W_{t}$ is normally distributed w.r.t. the measure $m^{\epsilon}(t,x)dx$ in ${\mathbb{R}}$ , we have

{\mathds{E}}[|W_{t}|^{\gamma}]=\int_{{\mathbb{R}}}|W_{t}|^{\gamma}m^{\epsilon}(t,x)dx=\frac{(2t)^{\gamma/2}}{\pi}\Gamma(\tfrac{\gamma+1}{2}),

where $\Gamma(\cdot)$ denotes the Gamma function. Integrating (5) w.r.t. $m^{\epsilon}(t,x)dx$ , using the previous formula, and recalling the initial condition for $m^{\epsilon}$ in (1.2), we obtain that $m^{\epsilon}$ satisfies

\displaystyle\int_{{\mathbb{R}}}|x|^{\gamma}m^{\epsilon}(t,x)dx

\displaystyle\leqslant 2^{\gamma-1}\int_{{\mathbb{R}}}|x|^{\gamma}m_{0}(x)dx+C_{1}^{\prime}+C_{2}^{\prime}\frac{(2T)^{\gamma/2}}{\pi}\Gamma(\tfrac{\zeta_{1}+1}{2}).

By Assumption 5, the right-hand side of the previous inequality is bounded independently of $\epsilon$ , for $0<\epsilon<1$ , as stated. ∎

Let $t\in[0,T]$ . Define $\beta^{\epsilon}_{t}\in{\mathcal{P}}({\mathbb{R}}^{2})$ by

\int_{{\mathbb{R}}^{2}}\psi(x,p)d\beta^{\epsilon}_{t}(x,p)=\int_{{\mathbb{R}}}\psi(x,\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,x))m^{\epsilon}(t,x){\rm d}x\quad\mbox{for all}~{}\psi\in C_{\bar{\zeta}}({\mathbb{R}}^{2}),

where $C_{\zeta}({\mathbb{R}}^{2})=\{\phi\in C({\mathbb{R}}^{2}):\lim_{|x|,|v|\to\infty}\frac{\phi(x,v)}{1+|x|^{\zeta_{1}}+|v|^{\zeta_{2}}}=0\}$ and $\bar{\zeta}=(\gamma_{1},\gamma_{2}^{\prime})$ . Note that the well definiteness of the measure $\beta^{\epsilon}_{t}$ is ensured by Proposition 5.2. Relying on the definition of $\beta^{\epsilon}_{t}$ , we define $\mu^{\epsilon}_{t}\in{\mathcal{P}}({\mathbb{R}}^{2})$ by

\int_{{\mathbb{R}}^{2}}\psi(x,-L_{v}(x,v))d\mu^{\epsilon}_{t}(x,v)=\int_{{\mathbb{R}}^{2}}\psi(x,p)d\beta^{\epsilon}_{t}(x,p)\quad\mbox{for all}~{}\psi\in C_{\bar{\zeta}}({\mathbb{R}}^{2}).

If Assumption 1 holds, the relation $v=-H_{p}(x,p)$ if and only if $p=-L_{v}(x,v)$ (see Remark 2.1), implies

\int_{{\mathbb{R}}^{2}}\psi(x,-H_{p}(x,p))d\beta^{\epsilon}_{t}(x,p)=\int_{{\mathbb{R}}^{2}}\psi(x,v)d\mu^{\epsilon}_{t}(x,v).

Finally, we define $\beta^{\epsilon},\mu^{\epsilon}\in\mathcal{U}^{\bar{\zeta}}\cap\mathcal{R}^{+}(\Omega)$ by

\int_{\Omega}f(t,x,v)d\beta^{\epsilon}(t,x,v)=\int_{0}^{T}\int_{{\mathbb{R}}^{2}}f(t,x,v)d\beta^{\epsilon}_{t}(x,v){\rm d}t,

and

\int_{\Omega}f(t,x,v)d\mu^{\epsilon}(t,x,v)=\int_{0}^{T}\int_{{\mathbb{R}}^{2}}f(t,x,v)d\mu^{\epsilon}_{t}(x,v){\rm d}t,

(5.9)

for all $f\in C_{\bar{\zeta}}(\Omega)$ (see Remark 3.4). Under Assumptions 1, 2, 4, 6, 7 and 8, the non-negative and finite Radon measures $\mu^{\epsilon}$ defined by (5.9) have a weak limit in $\mathcal{U}^{\bar{\zeta}}$ as $\epsilon\to 0$ .

We show the existence of a weak limit of the Radon measures $\mu^{\epsilon}$ defined by (5.9).

Proposition 5.3.

Suppose Assumptions 1, 2, 4-8 hold. Then, there exists $\mu\in\mathcal{U}^{\bar{\zeta}}\cap\mathcal{R}^{+}(\Omega)$ , where $\bar{\zeta}=(\gamma_{1},\gamma_{2}^{\prime})$ , such that, up to a sub-sequence, the sequence of Radon measures $\mu^{\epsilon}$ defined by (5.9) weakly converge to $\mu$ ; that is, for all $f\in C_{\bar{\zeta}}(\Omega)$

\begin{split}\int_{\Omega}f(t,x,v)d\mu^{\epsilon}(t,x,v)\to\int_{\Omega}f(t,x,v)d\mu(t,x,v).\end{split}

(5.10)

Proof.

By (5) and Proposition 5.2, we have

	$\displaystyle\int_{0}^{T}\int_{{\mathbb{R}}}\left(1+\|x\|^{\gamma}+\|H_{p}(x,\varpi^{\epsilon}(t)+u_{x}^{\epsilon}(t,x))\|^{\gamma_{2}^{\prime}+1}\right)m^{\epsilon}(t,x){\rm d}x{\rm d}t$
	$\displaystyle\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}\left(1+\|x\|^{\gamma}+\tilde{C}^{\gamma_{2}^{\prime}+1}\right)m^{\epsilon}(t,x){\rm d}x{\rm d}t$
	$\displaystyle\leqslant C(1+\tilde{C}^{\gamma_{2}^{\prime}+1}).$

Using the previous inequality, an argument similar to that in Remark 3.2 shows that the probability measures $\mu^{\epsilon}$ defined by (5.9) satisfy

		$\displaystyle\int_{\Omega}\left(1+\|x\|^{\gamma}+\|v\|^{\gamma_{2}^{\prime}+1}\right)d\mu^{\epsilon}(t,x,v)$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}^{2}}\left(1+\|x\|^{\gamma}+\|v\|^{\gamma_{2}^{\prime}+1}\right)d\mu^{\epsilon}_{t}(x,v){\rm d}t$
		$\displaystyle\leqslant C(1+\tilde{C}^{\gamma_{2}^{\prime}+1}),$		(5.11)

where $C$ and $\tilde{C}$ are independent of $\epsilon$ . Hence, $\mu^{\epsilon}\in\mathcal{U}^{\bar{\zeta}}\cap\mathcal{R}^{+}(\Omega)$ , with $\bar{\zeta}=(\gamma_{1},\gamma_{2}^{\prime})$ . Furthermore, (5) implies that the measure $\nu^{\epsilon}=\left(1+|x|^{\gamma_{1}}+|v|^{\gamma_{2}^{\prime}}\right)\mu^{\epsilon}(t,x,v)$ belongs to $\mathcal{R}^{+}(\Omega)$ and

\int_{\Omega}\left(t^{\alpha_{0}}+|x|^{\alpha_{0}}+|v|^{\alpha_{0}}\right)d\nu^{\epsilon}(t,x,v)<C,

where $0<\alpha_{0}<\min\{(\gamma-\gamma_{1}),\frac{1}{\gamma}(\gamma_{2}^{\prime}+1)(\gamma-\gamma_{1}),\frac{\gamma}{\gamma_{2}^{\prime}+1},1\}$ . Therefore, as $\epsilon\to 0$ , the sequence $\nu^{\epsilon}$ is tight ([38], Proposition 2.23). Hence, by Prohorov’s Theorem ([38], Theorem 2.29), there exists $\nu\in\mathcal{R}^{+}(\Omega)$ such that, up to a sub-sequence, which we still denote by $\nu^{\epsilon}$ , $\nu^{\epsilon}$ weakly converges to $\nu$ ; that is,

\int_{\Omega}\psi(t,x,v)d\nu^{\epsilon}(t,x,v)\to\int_{\Omega}\psi(t,x,v)d\nu(t,x,v)\quad\mbox{for all}\quad\psi\in C_{b}(\Omega).

(5.12)

Now, taking $\mu=\frac{1}{1+|x|^{\gamma_{1}}+|v|^{\gamma_{2}^{\prime}}}\nu$ , we notice that $\mu\in\mathcal{U}^{\bar{\zeta}}\cap\mathcal{R}^{+}(\Omega)$ . Moreover, recalling the definition of $\nu^{\epsilon}$ from (5.12), we deduce (5.10). ∎

Next, we show that the weak limit provided by Proposition 5.3 belongs to $\mathcal{H}(m_{0})$ .

Proposition 5.4.

Suppose Assumptions 1, 2, 4-8 hold. Let $\mu\in\mathcal{R}(\Omega)$ be such that, up to a sub-sequence, the Radon measures $\mu^{\epsilon}$ defined by (5.9) weakly converge to $\mu$ . Then, $\mu\in\mathcal{H}(m_{0})$ .

Proof.

The existence of $\mu$ is given by Proposition 5.3. By (5), we have that $\mu\in\mathcal{H}_{1}$ . Let $(u,m,\varpi)$ be the solution of (1.1) and (1.2) ([35], Theorem 1) . Let $\varphi\in C^{1}_{c}([0,T]\times{\mathbb{R}})$ . Because $m$ is a weak solution of the second equation in (1.1), we have

	$\displaystyle\int_{0}^{T}\int_{{\mathbb{R}}}\big{(}\varphi_{t}(t,x)-H_{p}(x,\varpi+u_{x})\varphi_{x}(t,x)\big{)}m(t,x)dx$
	$\displaystyle=\int_{{\mathbb{R}}}\varphi(T,x)m(T,x){\rm d}x-\int_{{\mathbb{R}}}\varphi(0,x)m_{0}(x){\rm d}x,$

and by (5.9)

	$\displaystyle\int_{\Omega}\varphi_{t}(t,x)+v\varphi_{x}(t,x)d\mu^{\epsilon}(t,x,v)$	$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}^{2}}\varphi_{t}(t,x)+v\varphi_{x}(t,x)~{}d\mu^{\epsilon}_{t}(x,v)dt$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}^{2}}\varphi_{t}(t,x)-H_{p}(x,p)\varphi_{x}(t,x)~{}d\beta^{\epsilon}_{t}(x,p)dt$
		$\displaystyle=\int_{0}^{T}\int_{{\mathbb{R}}}\varphi_{t}(t,x)-H_{p}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})\varphi_{x}(t,x)~{}m^{\epsilon}(t,x)dxdt.$

Now, taking into account (5) and arguing as in Remark 3.2, we deduce that the previous two identities also hold for any $\varphi\in\Lambda([0,T]\times{\mathbb{R}})$ . Hence, $\mu\in\mathcal{H}_{2}(m_{0},\nu)$ for $\nu=m(T,\cdot)\in{\mathcal{P}}({\mathbb{R}})$ . Finally, the third equation in (4.7) gives

\int_{\Omega}\eta(t)(v-Q(t))d\mu^{\epsilon}(t,x,v)=\int_{0}^{T}\int_{{\mathbb{R}}}\eta(t)(-H_{p}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})-Q(t))m^{\epsilon}(t,x)dxdt=0,

for all $\eta\in C([0,T])$ , which implies that $\mu\in\mathcal{H}_{3}$ . Therefore, $\mu\in\mathcal{H}(m_{0})$ as stated. ∎

Next, we prove the following technical lemma.

Lemma 5.5.

Let $x_{n}>y_{n}$ , $y_{n}\to+\infty$ and $\lim\limits_{n\to\infty}\frac{x_{n}}{y_{n}}=1$ . Suppose that $\phi\in L^{1}({\mathbb{R}})$ and $\int_{{\mathbb{R}}}|x|^{\sigma}\phi{\rm d}x<\infty$ for some $\sigma>0$ . Then,

\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\phi(x){\rm d}x{\rm d}y=\int_{{\mathbb{R}}}\phi(x){\rm d}x.

(5.13)

Proof.

After exchanging the order of the integrals on the left-hand side in (5.13), we have

\begin{split}&\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\phi(x){\rm d}x{\rm d}y=\int_{-2y_{n}}^{0}\phi(x)\int_{-y_{n}}^{x+y_{n}}{\rm d}y{\rm d}x\\ &+\int_{0}^{x_{n}-y_{n}}\phi(x)\int_{-y_{n}}^{x_{n}}{\rm d}y{\rm d}x+\int_{0}^{x_{n}-y_{n}}\phi(x)\int_{x+y_{n}}^{x_{n}}{\rm d}y{\rm d}x+\int_{x_{n}-y_{n}}^{2x_{n}}\phi(x)\int_{-x_{n}+x}^{x_{n}}{\rm d}y{\rm d}x\\ &=2x_{n}\int_{-2y_{n}}^{2x_{n}}\phi(x){\rm d}x+2(y_{n}-x_{n})\int_{-2y_{n}}^{0}\phi(x){\rm d}x+\int_{-2y_{n}}^{0}\phi(x)x{\rm d}x-\int_{0}^{2x_{n}}\phi(x)x{\rm d}x.\end{split}

Dividing the proceeding equation by $2x_{n}$ and letting $n\to\infty$ , we deduce (5.13). ∎

Now, relying on the previous results, we complete the second part of the proof of Theorem 1.3. This is the content of the following Lemma.

Lemma 5.6.

Suppose Assumptions 1-8 hold. Let $(u,m,\varpi)$ solve (1.1) and (1.2). Then

\int_{{\mathbb{R}}}\left(u(0,x)-u_{T}(x)\right)dm_{0}(x)-\int_{0}^{T}\varpi(t)Q(t)dt\geqslant\inf_{\begin{subarray}{c}\mu\in\mathcal{H}(m_{0})\end{subarray}}\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v).

Proof.

By Assumption 1 and (1.3), the following identity holds

L(x,v)=H_{p}(x,-L_{v}(x,v))(-L_{v}(x,v))-H(x,-L_{v}(x,v)).

(5.14)

Let $t\in[0,T]$ . By Remark 2.1 and (5), we have

\int_{{\mathbb{R}}}L(x,-H_{p}(x,\varpi^{\epsilon}+u_{x}^{\epsilon}))m^{\epsilon}(t,x)dx\leqslant\int_{{\mathbb{R}}}\left(C_{2}|x|^{\gamma_{1}}+C\right)m^{\epsilon}(t,x)dx,

where $C$ is independent of $x$ and $\epsilon$ . From the previous inequality, Assumption 5 and Proposition 5.2, and an argument similar to that in Remark 3.2, we get that the integral $\int_{{\mathbb{R}}^{2}}L(x,v)d\mu^{\epsilon}_{t}(x,v)$ exists and is finite. Hence, we integrate both sides of (5.14) w.r.t. $\mu^{\epsilon}_{t}$ , and we use the definition of $\beta_{t}^{\epsilon}$ to obtain

$\displaystyle\int_{{\mathbb{R}}^{2}}L(x,v)d\mu^{\epsilon}_{t}(x,v)$	$\displaystyle=\int_{{\mathbb{R}}^{2}}H_{p}(x,-L_{v}(x,v))(-L_{v}(x,v))-H(x,-L_{v}(x,v))d\mu^{\epsilon}_{t}(x,v)$
	$\displaystyle=\int_{{\mathbb{R}}^{2}}H_{p}(x,p)p-H(x,p)d\beta^{\epsilon}_{t}(x,p)$
	$\displaystyle=\int_{{\mathbb{R}}}\big{(}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})(\varpi^{\epsilon}+u^{\epsilon}_{x})-H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})\big{)}m^{\epsilon}dx.$	(5.15)

Let $a\in[-\frac{1}{2},0]$ be such that $|m^{\varepsilon}(t,a)|<\infty$ . By Proposition 5.2 and Assumption 5, we have that $\int_{{\mathbb{R}}}|x|m^{\epsilon}dx<\infty$ . Rewriting the first momentum of $m^{\varepsilon}$

\int_{{\mathbb{R}}}|x|m^{\epsilon}{\rm d}x=\int_{-a}^{0}|x|m^{\epsilon}{\rm d}x+\sum_{n=0}^{\infty}\int_{n}^{n+1}xm^{\epsilon}{\rm d}x+\int_{-n-a-1}^{-n-a}|x|m^{\epsilon}dx<\infty,

we deduce that there exists $N_{0}$ such that for all $N\geqslant N_{0}$

\int_{N}^{N+1}xm^{\epsilon}dx+\int_{-N-a-1}^{-N-a}|x|m^{\epsilon}{\rm d}x=\int_{N}^{N+1}xm^{\epsilon}dx+\int_{-N-1}^{-N}|x-a|m^{\epsilon}(t,x-a){\rm d}x\leqslant\frac{C}{N}.

The previous estimates with Chebyshev’s inequality imply

\begin{split}\bigg{|}\{x\in[N,N+1]:\quad xm^{\epsilon}>\frac{1}{\sqrt{N}}\}\bigg{|}\leqslant\sqrt{N}\int_{N}^{N+1}xm^{\epsilon}dx\leqslant\frac{C}{\sqrt{N}},\\ \bigg{|}\{x\in[-N-1,-N]:\quad|x-a|m^{\epsilon}(t,x-a)>\frac{1}{\sqrt{N}}\}\bigg{|}\leqslant\sqrt{N}\int_{-N-a-1}^{-N-a}|x|m^{\epsilon}dx\leqslant\frac{C}{\sqrt{N}}.\end{split}

Because $a\in[-\frac{1}{2},0]$ , there exists a sequence $\{x_{n}\}$ such that

\begin{split}&x_{n}\geqslant 0,\quad\lim\limits_{n\to\infty}x_{n}=+\infty,\\ &\lim\limits_{n\to\infty}x_{n}m^{\epsilon}(t,2x_{n})=\lim\limits_{n\to\infty}x_{n}m^{\epsilon}(t,-2x_{n}-2a)=0.\\ \end{split}

(5.16)

Let $y_{n}=x_{n}+a$ .

By Assumption 5 follows that there exists $\sigma>0$ such that $\int_{{\mathbb{R}}}|x|^{\gamma_{1}+\sigma}m_{0}{\rm d}x<\infty$ . Then, relying on Proposition 5.2 and using Lemma 5.5, we rewrite (5)

\begin{split}&\int_{{\mathbb{R}}^{2}}L(x,v)d\mu^{\epsilon}_{t}(x,v)=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\big{(}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})(\varpi^{\epsilon}+u^{\epsilon}_{x})\big{.}\\ &\big{.}-H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})\big{)}m^{\epsilon}{\rm d}x{\rm d}y=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}\big{(}u^{\epsilon}-u_{T}\big{)}_{x}\\ &+H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}m^{\epsilon}-H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}{\rm d}x{\rm d}y.\end{split}

(5.17)

Because $H$ is separable, $u^{\epsilon}(t,\cdot),u_{T}\in\mbox{Lip}({\mathbb{R}})$ , from Proposition 5.2, we deduce

\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{-y_{n}}^{x_{n}}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}\big{(}u^{\epsilon}-u_{T}\big{)}\bigg{|}_{-y_{n}+y}^{x_{n}+y}{\rm d}y\right|\leqslant\lim\limits_{n\to\infty}\frac{C}{x_{n}}\int_{{\mathbb{R}}}|x|m^{\epsilon}{\rm d}x=0.

Integrating by parts the first term on the right-hand side in (5.17), using the preceding equality, and the definition of $\beta^{\epsilon}_{t}$ , (5.17) becomes

		$\displaystyle\int_{{\mathbb{R}}^{2}}L(x,v)d\mu^{\epsilon}_{t}(x,v)$
		$\displaystyle=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}-\big{(}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}\big{)}_{x}\big{(}u^{\epsilon}-u_{T}\big{)}-H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}{\rm d}x{\rm d}y$
		$\displaystyle+\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\int_{\mathbb{R}}H_{p}(x,p)\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\beta^{\epsilon}_{t}(x,p){\rm d}y.$		(5.18)

Next, we prove well definiteness of several integrals. Note that Assumption 5 and the definition of $\mu^{\epsilon}_{t}$ , yield

\begin{split}\left|\int_{{\mathbb{R}}^{2}}v|x|^{\sigma}\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}_{t}(x,v)\right|&=\left|\int_{{\mathbb{R}}^{2}}H_{p}(x,p)|x|^{\sigma}\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\beta^{\epsilon}_{t}(x,p)\right|\\ &=\left|\int_{{\mathbb{R}}}H_{p}(x,\varpi^{\epsilon}+u_{x}^{\epsilon})|x|^{\sigma}\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}m^{\epsilon}{\rm d}x\right|\leqslant C,\end{split}

for $\gamma_{1}<\gamma_{1}+\sigma<\gamma$ . Relying on the preceding estimate and considering Lemma 5.5, we obtain

-\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\int_{\mathbb{R}}v\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}_{t}(x,v)=-\int_{{\mathbb{R}}^{2}}v\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}_{t}(x,v).

(5.19)

By the second-order energy estimate in (4.12) and using Young’s inequality, we have

\begin{split}\int_{0}^{T}\int_{{\mathbb{R}}}|u_{xx}^{\epsilon}||x|^{\frac{\gamma}{2}}m^{\epsilon}{\rm d}x{\rm d}t\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}(u_{xx}^{\epsilon})^{2}m^{\epsilon}+|x|^{\gamma}m^{\epsilon}{\rm d}x{\rm d}t\leqslant C.\end{split}

Hence, Lemma 5.5 implies that

\begin{split}\int_{0}^{T}\int_{{\mathbb{R}}}|u_{xx}^{\epsilon}|m^{\epsilon}{\rm d}x{\rm d}t=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}|u_{xx}^{\epsilon}|m^{\epsilon}{\rm d}x{\rm d}t\leqslant C.\end{split}

Using the previous estimate and taking into account that $u^{\epsilon}\in\mbox{Lip}({\mathbb{R}})$ for all $t\in[0,T]$ , we get

\begin{split}&\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}m_{x}^{\epsilon}u^{\epsilon}_{x}{\rm d}x{\rm d}y{\rm d}t\right|\leqslant\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{0}^{T}\int_{-y_{n}}^{x_{n}}m^{\epsilon}(t,x_{n}+y)u^{\epsilon}_{x}(t,x_{n}+y)\right.\\ &\left.-m^{\epsilon}(t,-y_{n}+y)u^{\epsilon}_{x}(t,-y_{n}+y){\rm d}y{\rm d}t\bigg{|}\right.+\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}|u_{xx}^{\epsilon}|m^{\epsilon}{\rm d}x{\rm d}t\\ &\leqslant\lim\limits_{n\to\infty}\frac{1}{x_{n}}\int_{0}^{T}\int_{{\mathbb{R}}}m^{\epsilon}|u_{x}^{\epsilon}|{\rm d}y{\rm d}t+\int_{0}^{T}\int_{{\mathbb{R}}}|u_{xx}^{\epsilon}|m^{\epsilon}{\rm d}y{\rm d}t\leqslant C.\end{split}

(5.20)

Because $|m^{\varepsilon}(t,a)|<\infty$ from (5.16), we have

\begin{split}&\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}u^{\epsilon}\bigg{|}_{-y_{n}+y}^{x_{n}+y}{\rm d}y\right|\\ &\leqslant\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}(t,x_{n}+y)u^{\epsilon}(t,x_{n}+y){\rm d}y-\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}(t,-y_{n}+y)u^{\epsilon}(t,-y_{n}+y){\rm d}y\right|\\ &=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left|\int_{x_{n}-y_{n}}^{2x_{n}}m^{\epsilon}_{x}(t,y)u^{\epsilon}(t,y){\rm d}y-\int_{-2y_{n}}^{x_{n}-y_{n}}m^{\epsilon}_{x}(t,y)u^{\epsilon}(t,y){\rm d}y\right|\\ &\leqslant\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left(m^{\epsilon}(t,2x_{n})|u^{\epsilon}(t,2x_{n})|+m^{\epsilon}(t,-2y_{n})|u^{\epsilon}(t,-2y_{n})|+2m^{\epsilon}(t,a)|u^{\epsilon}(t,a)|\right.\\ &+2\int_{{\mathbb{R}}}m^{\epsilon}|u_{x}^{\epsilon}|{\rm d}x)=0.\end{split}

(5.21)

Furthermore, (5.20) and (5.21), yield

\begin{split}\left|\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}m^{\epsilon}_{xx}u^{\epsilon}{\rm d}x{\rm d}y{\rm d}t\right|&\leqslant\left|\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}u^{\epsilon}\bigg{|}_{-y_{n}+y}^{x_{n}+y}{\rm d}y{\rm d}t\right|\\ &+\left|\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}m^{\epsilon}_{x}u^{\epsilon}_{x}{\rm d}x{\rm d}y{\rm d}t\right|\leqslant C.\end{split}

(5.22)

Note that (5.20), (5.21) and (5.22) also hold for $u_{T}$ .

Because $\varpi^{\epsilon}\in C[0,T]$ , then $H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})\in L^{\infty}_{loc}([0,T]\times{\mathbb{R}})$ , which with the regularity of heat equation implies that $u_{x}\in C([0,T]\times{\mathbb{R}})$ and $u_{xx}\in L_{loc}^{p}([0,T]\times{\mathbb{R}})$ for every $p\in[1,\infty)$ . Therefore, the second and the first equation in (4.7) imply

	$\displaystyle-\big{(}H_{p}(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}\big{)}_{x}\big{(}u^{\epsilon}-u_{T}\big{)}=\left(\epsilon m^{\epsilon}_{xx}-m^{\epsilon}_{t}\right)\big{(}u^{\epsilon}-u_{T}\big{)},$
	$\displaystyle-H(x,\varpi^{\epsilon}+u^{\epsilon}_{x})m^{\epsilon}=-\left(\epsilon u^{\epsilon}_{xx}+u^{\epsilon}_{t}\right)m^{\epsilon}.$

Relying on (5.22) and using the preceding identities and the identities in (5.19) after integrating on $[0,T]$ the equation in (5), we obtain

\begin{split}&\int_{\Omega}L(x,v)d\mu^{\epsilon}_{t}(x,v){\rm d}t=-\int_{0}^{T}\int_{{\mathbb{R}}^{2}}v\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}_{t}(x,v){\rm d}t\\ &+\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}-m^{\epsilon}_{t}\big{(}u^{\epsilon}-u_{T}\big{)}+\epsilon m^{\epsilon}_{xx}\big{(}u^{\epsilon}-u_{T}\big{)}-u^{\epsilon}_{t}m^{\epsilon}-\epsilon u^{\epsilon}_{xx}m^{\epsilon}{\rm d}x{\rm d}y{\rm d}t\\ &=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}-\big{(}m^{\epsilon}\big{(}u^{\epsilon}-u_{T}\big{)}\big{)}_{t}+\epsilon m^{\epsilon}_{xx}\big{(}u^{\epsilon}-u_{T}\big{)}-\epsilon u^{\epsilon}_{xx}m^{\epsilon}{\rm d}x{\rm d}y{\rm d}t\\ &-\int_{\Omega}v\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}_{t}(x,v){\rm d}t.\end{split}

(5.23)

Taking into account (5.21) and integrating by parts, we have

		$\displaystyle\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left\|\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}m^{\epsilon}_{xx}u^{\epsilon}-u^{\epsilon}_{xx}m^{\epsilon}{\rm d}x{\rm d}y{\rm d}t\right\|$
		$\displaystyle=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left\|\int_{0}^{T}\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}u^{\epsilon}-u^{\epsilon}_{x}m^{\epsilon}\bigg{\|}_{-y_{n}+y}^{x_{n}+y}{\rm d}y{\rm d}t\right\|$
		$\displaystyle\leqslant\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\left(\left\|\int_{0}^{T}\int_{-y_{n}}^{x_{n}}m^{\epsilon}_{x}u^{\epsilon}\bigg{\|}_{-y_{n}+y}^{x_{n}+y}{\rm d}y{\rm d}t\right\|+C\int_{0}^{T}\int_{{\mathbb{R}}}m^{\epsilon}{\rm d}x{\rm d}t\right)$
		$\displaystyle=0.$		(5.24)

Thus, recalling the definition of $\mu^{\epsilon}$ , (5.9), and by using (5.22), (5) in (5.23), we get

	$\displaystyle\int_{\Omega}L(x,v)d\mu^{\epsilon}(t,x,v)$	$\displaystyle=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}-\left(m^{\epsilon}\left(u^{\epsilon}-u_{T}\right)\right)_{t}-\epsilon m^{\epsilon}_{xx}u_{T}{\rm d}x{\rm d}y{\rm d}t$
		$\displaystyle-\int_{\Omega}v\big{(}\varpi^{\epsilon}+u_{T}^{\prime}\big{)}d\mu^{\epsilon}(t,x,v).$

After rearranging the terms in the previous equation, we obtain

		$\displaystyle\int_{\Omega}L(x,v)+vu_{T}^{\prime}d\mu^{\epsilon}(t,x,v)=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}\big{(}u^{\epsilon}(0,x)-u_{T}(x)\big{)}m_{0}{\rm d}x{\rm d}y{\rm d}t$
		$\displaystyle-\epsilon\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}m^{\epsilon}_{xx}u_{T}{\rm d}x{\rm d}y{\rm d}t-\int_{\Omega}v\varpi^{\epsilon}d\mu^{\epsilon}(t,x,v).$		(5.25)

Now, we pass to the limit in (5) as follows. By Assumptions 1, 4, 6 and 7, Theorem 1 in [35] guarantees the existence of a sequence such that $u^{\epsilon}\to u$ and $\varpi^{\epsilon}\to\varpi$ uniformly, where, for $\varpi$ , $u$ solves the first equation in (1.1) in the viscosity sense. Furthermore, by Proposition 4.2, $\varpi\in W^{1,\infty}([0,T])$ . Remark 1 implies that $L(x,v)+vu_{T}^{\prime}(x)$ belongs to $C_{\bar{\zeta}}(\Omega)$ , therefore extracting a further sub-sequence out of the previous sequence, Proposition 5.3 gives the existence of a weak limit $\mu\in\mathcal{U}^{\bar{\zeta}}\cap\mathcal{R}(\Omega)$ for $\mu^{\epsilon}$ and (5.10) holds for $L(x,v)+vu_{T}^{\prime}(x)$ . Using these, by letting $\epsilon\to 0$ in (5), we obtain

	$\displaystyle\int_{0}^{T}\int_{{\mathbb{R}}^{2}}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v)$	$\displaystyle=\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}(u(0,x)-u_{T}(x))m_{0}{\rm d}x{\rm d}y$
		$\displaystyle-\int_{\Omega}v\varpi(t)d\mu(t,x,v).$		(5.26)

Furthermore, by Proposition 4.2, $\varpi\in W^{1,\infty}([0,T])$ , and by Proposition 5.4 $\mu\in\mathcal{H}(m_{0})$ . In particular, $\mu\in\mathcal{H}_{3}$ . Therefore,

\int_{0}^{T}\int_{{\mathbb{R}}^{2}}v\varpi(t)d\mu(t,x,v)=\int_{0}^{T}Q(t)\varpi(t){\rm d}t.

By Assumption 5 and Lemma 5.5, we deduce that

\lim\limits_{n\to\infty}\frac{1}{2x_{n}}\int_{0}^{T}\int_{-y_{n}}^{x_{n}}\int_{-y_{n}+y}^{x_{n}+y}(u(0,x)-u_{T}(x))m_{0}{\rm d}x{\rm d}y=\int_{0}^{T}\int_{{\mathbb{R}}}u(0,x)-u_{T}(x)dm_{0}(x).

Therefore, from (5), we obtain

\displaystyle\int_{\Omega}L(x,v)+vu_{T}^{\prime}(x)d\mu(t,x,v)

\displaystyle=\int_{{\mathbb{R}}}u(0,x)-u_{T}(x)dm_{0}(x)-\int_{0}^{T}Q(t)\varpi(t)dt,

which completes the proof. ∎

References

[1] C. Alasseur, I. Ben Taher, and A. Matoussi. An extended mean field game for storage in smart grids. Journal of Optimization Theory and Applications, 184(2):644–670, 2020.
[2] A. Alharbi, T. Bakaryan, R. Cabral, S. Campi, N. Christoffersen, P. Colusso, O. Costa, S. Duisembay, R. Ferreira, D. Gomes, S. Guo, J. Gutierrez, P. Havor, M. Mascherpa, S. Portaro, R. Ricardo de Lima, F. Rodriguez, J. Ruiz, F. Saleh, S. Calum, T. Tada, X. Yang, and Z. Wróblewska. A price model with finitely many agents. Bulletin of the Portuguese Mathematical Society, 2019.
[3] T. Basar and R. Srikant. Revenue-maximizing pricing and capacity expansion in a many-users regime. In Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, volume 1, pages 294–301 vol.1, 2002.
[4] H. H. Bauschke, P. L. Combettes, et al. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer International Publishing, 2 edition, 2017.
[5] A. Biryuk and D. Gomes. An introduction to the Aubry-Mather theory. São Paulo J. Math. Sci., 4(1):17–63, 2010.
[6] M. Burger, L. A. Caffarelli, P. A. Markowich, and M.-T. Wolfram. On the asymptotic behavior of a Boltzmann-type price formation model. Commun. Math. Sci., 12(7):1353–1361, 2014.
[7] L. A. Caffarelli, P. A. Markowich, and J.-F. Pietschmann. On a price formation free boundary model by Lasry and Lions. C. R. Math. Acad. Sci. Paris, 349(11-12):621–624, 2011.
[8] L. A. Caffarelli, P. A. Markowich, and M.-T. Wolfram. On a price formation free boundary model by Lasry and Lions: the Neumann problem. C. R. Math. Acad. Sci. Paris, 349(15-16):841–844, 2011.
[9] F. Cagnetti, D. Gomes, H. Mitake, and H. Tran. A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians. Ann. Inst. H. Poincaré Anal. Non Linéaire, 32(1):183–200, 2015.
[10] P. Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004.
[11] Piermarco Cannarsa, Wei Cheng, Cristian Mendico, and Kaizhi Wang. Weak kam approach to first-order mean field games with state constraints, 2020.
[12] P. Cardaliaguet. Long time average of first order mean field games and weak KAM theory. Dyn. Games Appl., 3(4):473–488, 2013.
[13] P. Cardaliaguet. A short course on mean field games. 2018.
[14] B. Djehiche, J. Barreiro-Gomez, and H. Tembine. Price Dynamics for Electricity in Smart Grid Via Mean-Field-Type Games. Dynamic Games and Applications, 10(4):798–818, December 2020.
[15] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions, Revised Edition. Textbooks in Mathematics. CRC Press, 2015.
[16] L. C. Evans and D. Gomes. Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal., 157(1):1–33, 2001.
[17] L. C. Evans and D. Gomes. Effective Hamiltonians and averaging for Hamiltonian dynamics II. Archive for rational mechanics and analysis, 161(4):271–305, 2002.
[18] A. Fathi. Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 325(6):649–652, 1997.
[19] A. Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math., 324(9):1043–1046, 1997.
[20] A. Fathi. Orbite hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 326:1213–1216, 1998.
[21] A. Fathi. Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math., 327:267–270, 1998.
[22] O. Féron, P. Tankov, and L. Tinsi. Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player. Risks, 8(4):1–1, December 2020.
[23] W. Fleming and D. Vermes. Convex duality approach to the optimal control of diffusions. SIAM J. Control Optim., 27(5):1136–1155, 1989.
[24] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts. Wiley-Interscience, 2 edition, 1999.
[25] I. Fonseca and G. Leoni. Modern methods in the calculus of variations: $L^{p}$ spaces. Springer Monographs in Mathematics. Springer, New York, 2007.
[26] M. Fujii and A. Takahashi. A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition. Papers 2003.03035, arXiv.org, March 2020.
[27] M. Fujii and A. Takahashi. Equilibrium price formation with a major player and its mean field limit, 2021.
[28] G. Pólya G. H. Hardy, J. E. Littlewood. Inequalities. Cambridge University Press, 2 edition, 1934.
[29] D. Gomes. A stochastic analogue of Aubry-Mather theory. Nonlinearity, 15(3):581–603, 2002.
[30] D. Gomes. Duality principles for fully nonlinear elliptic equations. In Trends in partial differential equations of mathematical physics, volume 61 of Progr. Nonlinear Differential Equations Appl., pages 125–136. Birkhäuser, Basel, 2005.
[31] D. Gomes. Generalized Mather problem and selection principles for viscosity solutions and Mather measures. Adv. Calc. Var., 1(3):291–307, 2008.
[32] D. Gomes, J. Gutierrez, and R. Ribeiro. A mean field game price model with noise. Math. Eng., 3(4):Paper No. 028, 14, 2021.
[33] D. Gomes, H. Mitake, and K. Terai. The selection problem for some first-order stationary mean-field games. Netw. Heterog. Media, 15(4):681–710, 2020.
[34] D. Gomes, H. Mitake, and H. Tran. The selection problem for discounted Hamilton-Jacobi equations: some non-convex cases. J. Math. Soc. Japan, 70(1):345–364, 2018.
[35] D. Gomes and J. Saúde. A Mean-Field Game Approach to Price Formation. Dyn. Games Appl., 11(1):29–53, 2021.
[36] D. Gomes and E. Valdinoci. Duality theory, representation formulas and uniqueness results for viscosity solutions of Hamilton-Jacobi equations. In Dynamics, games and science. II, volume 2 of Springer Proc. Math., pages 361–386. Springer, Heidelberg, 2011.
[37] Diogo A. Gomes, Hiroyoshi Mitake, and Hung V. Tran. The large time profile for hamilton–jacobi–bellman equations, 2020.
[38] Werner Kirsch. A survey on the method of moments. 2015.
[39] A. Lachapelle, J.-M. Lasry, C.-A. Lehalle, and P.-L. Lions. Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis. Mathematics and Financial Economics, 10(3):223–262, 2016.
[40] R. M. Lewis and R. B. Vinter. Relaxation of optimal control problems to equivalent convex programs. J. Math. Anal. Appl., 74(2):475–493, 1980.
[41] R. Mañé. On the minimizing measures of Lagrangian dynamical systems. Nonlinearity, 5(3):623–638, 1992.
[42] P. A. Markowich, N. Matevosyan, J.-F. Pietschmann, and M.-T. Wolfram. On a parabolic free boundary equation modeling price formation. Math. Models Methods Appl. Sci., 19(10):1929–1957, 2009.
[43] Anis Matoussi, Clémence Alasseur, and Imen Ben Taher. An extended mean field game for storage in smart grids. 2018.
[44] H. Mitake and H. Tran. Selection problems for a discount degenerate viscous Hamilton-Jacobi equation. Adv. Math., 306:684–703, 2017.
[45] Hung V. Tran Nam Q. Le, Hiroyoshi Mitake. Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations: VIASM 2016. Lecture Notes in Mathematics 2183. Springer International Publishing, 1 edition, 2017.
[46] H. Shen and T. Basar. Pricing under information asymmetry for a large population of users. Telecommun. Syst., 47(1-2):123–136, 2011.
[47] A. Shrivats, D. Firoozi, and S. Jaimungal. A Mean-Field Game Approach to Equilibrium Pricing, Optimal Generation, and Trading in Solar Renewable Energy Certificate Markets. Papers 2003.04938, arXiv.org, March 2020.
[48] C. Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.

	$\displaystyle\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\|\varphi^{n}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi(t,x)\theta\left(\tfrac{x}{n}\right)\right\|\leqslant Cn,$		(3.8)
	$\displaystyle\max_{(t,x,v)\in\Omega}\|\phi^{n}_{t}(t,x,v)\|\leqslant\max_{(t,x)\in\Omega}\|\varphi^{n}_{t}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi_{t}(t,x)\theta\left(\tfrac{x}{n}\right)\right\|\leqslant C,$
	$\displaystyle\|\|\phi^{n}_{x}\|\|_{C(\Omega)}\leqslant\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\|\varphi^{n}_{x}(t,x)\|=\max_{(t,x)\in[0,T]\times{\mathbb{R}}}\left\|\varphi_{x}(t,x)\theta\left(\tfrac{x}{n}\right)+\tfrac{1}{n}\varphi(t,x)\theta^{\prime}\left(\tfrac{x}{n}\right)\right\|\leqslant C.$

		$\displaystyle\left\|\int_{0}^{T}\int_{n<\|x\|<2n}\int_{n<\|v\|<2n}\phi^{n}_{t}+v\phi^{n}_{x}d\mu(t,x,v)\right\|$
		$\displaystyle\leqslant\int_{0}^{T}\int_{n<\|x\|<2n}\int_{n<\|v\|<2n}C(\|v\|+1)d\mu(t,x,v)$
		$\displaystyle\leqslant\int_{\Omega}C(\|v\|+1)\left(\theta\left(\frac{2x}{n}-3\right)+\theta\left(\frac{2x}{n}+3\right)\right)\left(\theta\left(\frac{2v}{n}-3\right)+\theta\left(\frac{2v}{n}+3\right)\right)d\mu(t,x,v)$
		$\displaystyle\leqslant\int_{0}^{T}\int_{{\mathbb{R}}}2C(\|Q(t)\|+1)\left(\theta\left(\frac{2(x+\overline{X}(t))}{n}-3\right)+\theta\left(\frac{2(x+\overline{X}(t))}{n}+3\right)\right)dm_{0}(x)$
		$\displaystyle\leqslant TC\int_{\frac{n}{2}-\overline{M}<\|x\|<\frac{5}{2}n+\overline{M}}dm_{0}(x)=o(1).$		(3.9)

		$\displaystyle\left\|\left(\int_{n}^{2n-\overline{X}(T)}+\int_{-2n-\overline{X}(T)}^{-n}\right)\varphi^{n}(T,x+\overline{X}(T))dm_{0}(x)\right\|$
		$\displaystyle\leqslant\int_{n-\overline{M}<\|x\|<2n+\overline{M}}\left\|\varphi^{n}(T,x+\overline{X}(T))\right\|dm_{0}(x)\leqslant C\int_{n-\overline{M}<\|x\|<2n+\overline{M}}\|x\|dm_{0}(x)=o(1),$
		$\displaystyle\left\|\int_{n<\|x\|<2n}\varphi^{n}(0,x)dm_{0}(x)\right\|\leqslant C\int_{n<\|x\|<2n}\|x\|dm_{0}(x)=o(1),$		(3.10)

	$\displaystyle\|q(t,x;s,y)\|$	$\displaystyle\leqslant\left\|H\big{(}x,\varpi(t)+w_{x}(t-s,x-y))-H(x,\varpi(t-s)+w_{x}(t-s,x-y)\big{)}\right\|$
		$\displaystyle+\left\|H\big{(}x,\varpi(t-s)+w_{x}(t-s,x-y))-H(x-y,\varpi(t-s)+w_{x}(t-s,x-y)\big{)}\right\|$
		$\displaystyle\leqslant C\|\varpi(t)-\varpi(t-s)\|\left(\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}-1}+1\right)$
		$\displaystyle+C\|y\|\big{(}\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}}+1\big{)}$
		$\displaystyle\leqslant C^{\prime}s\left(\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}-1}+1\right)$
		$\displaystyle+C\|y\|\big{(}\left\|\varpi(t-s)+w_{x}(t-s,x-y)\right\|^{\gamma_{2}}+1\big{)}$
		$\displaystyle\leqslant C^{\prime}(s+\|y\|),$

	$\displaystyle\|\mathrm{x}_{t}\|^{\gamma}$	$\displaystyle\leqslant 2^{{}^{\gamma}-1}\left(\|x\|^{\gamma}+2^{\gamma-1}\left(T^{\gamma}\tilde{C}^{\gamma}+\sqrt{2\epsilon}^{\gamma}\|W_{t}\|^{\gamma}\right)\right)$
		$\displaystyle\leqslant 2^{{}^{\gamma}-1}\|x\|^{\gamma}+C_{1}^{\prime}+C_{2}^{\prime}\|W_{t}\|^{\gamma},$		(5.8)