A Quadratic Mean Field Games Model for the Langevin Equation

Abstract

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.

1. Introduction

The Mean Field Games (MFG in short) theory concerns the study of differential games with a large number of rational, indistinguishable agents and the characterization of the corresponding Nash equilibria. In the original model introduced in [1,2], an agent can typically act on its velocity (or other first order dynamical quantities) via a control variable. Mean Field Games where agents control the acceleration have been recently proposed in [3,4,5].

A prototype of stochastic process involving acceleration is given by the Langevin diffusion process, which can be formally defined as

$$\ddot{X}\left(t\right)=-b\left(X\left(t\right)\right)+\sigma \dot{B}\left(t\right),$$

(1)

where $\ddot{X}$ is the second time derivative of the stochastic process X, B a Brownian motion and $\sigma$ a positive parameter. The solution of (1) can be rewritten as a Markov process $(X,V)$ solving

$$\left\{\begin{array}{c}\dot{X}\left(t\right)=V\left(t\right),\hfill \\ \dot{V}\left(t\right)=-b\left(X\left(t\right)\right)+\sigma \dot{B}\left(t\right).\hfill \end{array}\right.$$

The probability density function of the previous process satisfies the kinetic Fokker–Planck equation

$${\partial }_{t}p-\frac{{\sigma }^2}{2}{\Delta }_{v}p-b\left(x\right)·D_{v}p+v·D_{x}p=0\mathrm{in}(0,\infty )\times {\mathbb{R}}^d\times {\mathbb{R}}^d.$$

The previous equation, in the case $b\equiv 0$, was first studied by Kolmogorov [6] who provided an explicit formula for its fundamental solution. Then considered by Hörmander [7] as motivating example for the general theory of the hypoelliptic operators (see also [8,9,10]).

We consider a Mean Field Games model where the dynamics of the single agent is given by a controlled Langevin diffusion process, i.e.,

$$\left\{\begin{array}{cc}\dot{X}\left(s\right)=V\left(s\right),\hfill & s\ge t\hfill \\ \dot{V}\left(s\right)=-b\left(X\left(s\right)\right)+\alpha \left(s\right)+\sigma \dot{B}\left(s\right)\hfill & s\ge t\hfill \\ X\left(t\right)=x,V\left(t\right)=v\hfill \end{array}\right.$$

(2)

for $(t,x,v)\in [0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d$. In (2), the control law $\alpha :[t,T]\to {\mathbb{R}}^d$, which is a progressively measurable process with respect to a fixed filtered probability space such that $\mathbb{E}[{\int }_t^T|\alpha \left(t\right){|}^{2}dt]<+\infty$, is chosen to maximize the functional

$$\begin{array}{cc}\hfill J(t,x,v;\alpha )={\mathbb{E}}_{t,(x,v)}\{{\int }_t^T& \left[f(X\left(s\right),V\left(s\right),m\left(s\right))-\frac{1}{2}{\left|\alpha \left(s\right)\right|}^2\right]ds\hfill \\ \hfill & +u_T(X\left(T\right),V\left(T\right))\},\hfill \end{array}$$

where $m\left(s\right)$ is the distribution of the agents at time s. Let u the value function associated with the previous control problem, i.e.,

$$u(t,x,v)=\underset{\alpha \in {\mathcal{A}}_t}{sup}\left\{J(t,x,v;\alpha )\right\}$$

where ${\mathcal{A}}_t$ is the the set of the control laws. Formally, the couple $(u,m)$ satisfies the MFG system (see Section 4.1 in [3] for more details)

$$\left\{\begin{array}{c}{\partial }_{t}u+\frac{{\sigma }^2}{2}{\Delta }_{v}u-b\left(x\right)·D_{v}u+v·D_{x}u+\frac{1}{2}{\left|D_{v}u\right|}^2=-f(x,v,m)\hfill \\ {\partial }_{t}m-\frac{{\sigma }^2}{2}{\Delta }_{v}m-b\left(x\right)·D_{v}m+v·D_{x}m+{\mathrm{div}}_v\left(m{D}_{v}u\right)=0\hfill \\ m(0,x,v)=m_0(x,v),u(T,x,v)=u_T(x,v).\hfill \end{array}\right.$$

(3)

for $(t,x,v)\in (0,T)\times {\mathbb{R}}^d\times {\mathbb{R}}^d$. The first equation is a backward Hamilton–Jacobi–Bellman equation, degenerate in the x-variable and with a quadratic Hamiltonian in the v variable, and the second equation is forward kinetic Fokker–Planck equation. In the standard setting, MFG systems with quadratic Hamiltonians has been extensively considered in literature both as a reference model for the general theory and also since, thanks to the Hopf-Cole change of variable, the nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation, allowing to use all the tools developed for this type of problem (see for example [2,11,12,13,14,15]). Recently, a similar procedure has been used for ergodic hypoelliptic MFG with quadratic cost in [16] and for a flocking model involving kinetic equations in Section 4.7.3 of [17].

We study (3) by means of a change of variable introduced in [11,14] for the standard case. By defining the new unknowns $\varphi =e^{u/{\sigma }^2}$ and $\psi =m{e}^{-u/{\sigma }^2}$, the system (3) is transformed into a system of two kinetic Fokker–Planck equations

$$\left\{\begin{array}{c}{\partial }_t\varphi +\frac{{\sigma }^2}{2}{\Delta }_v\varphi -b\left(x\right)·D_v\varphi +v·D_x\varphi =-\frac{1}{{\sigma }^2}f(x,v,\psi \varphi )\varphi \hfill \\ {\partial }_t\psi -\frac{{\sigma }^2}{2}{\Delta }_v\psi -b\left(x\right)·D_v\psi +v·D_x\psi =\frac{1}{{\sigma }^2}f(x,v,\psi \varphi )\psi \hfill \\ \psi (0,x,v)=\frac{m_0(x,v)}{\varphi (0,x,v)},\varphi (T,x,v)=e^{\frac{u_T(x,v)}{{\sigma }^2}}.\hfill \end{array}\right.$$

(4)

for $(t,x,v)\in (0,T)\times {\mathbb{R}}^d\times {\mathbb{R}}^d$. In the previous problem, the coupling between the two equations is only in the source terms. Following [14], we prove existence of a weak solution to (4) by showing the convergence of an iterative scheme defined, starting from ${\psi }^{\left(0\right)}\equiv 0$, by solving alternatively the backward problem

$$\left\{\begin{array}{cc}{\partial }_t{\varphi }^{(k+\frac{1}{2})}+\frac{{\sigma }^2}{2}{\Delta }_v{\varphi }^{(k+\frac{1}{2})}\hfill & -b\left(x\right)·D_v{\varphi }^{(k+\frac{1}{2})}+v·D_x{\varphi }^{(k+\frac{1}{2})}\hfill \\ \hfill & =-\frac{1}{{\sigma }^2}f({\psi }^{\left(k\right)}{\varphi }^{(k+\frac{1}{2})}){\varphi }^{(k+\frac{1}{2})}\hfill \\ {\varphi }^{(k+\frac{1}{2})}(T,x,v)=e^{\frac{u_T(x,v)}{{\sigma }^2}},\hfill \end{array}\right.$$

(5)

and the forward one

$$\left\{\begin{array}{cc}{\partial }_t{\psi }^{(k+1)}-\frac{{\sigma }^2}{2}{\Delta }_v{\psi }^{(k+1)}\hfill & -b\left(x\right)·D_v{\psi }^{(k+1)}+v·D_x{\psi }^{(k+1)}\hfill \\ \hfill & =\frac{1}{{\sigma }^2}f({\psi }^{(k+1)}{\varphi }^{(k+\frac{1}{2})}){\psi }^{(k+1)}\hfill \\ {\psi }^{(k+1)}(0,x,v)=\frac{m_0(x,v)}{{\varphi }^{(k+\frac{1}{2})}(0,x,v)}.\hfill \end{array}\right.$$

(6)

We show that the resulting sequence $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$, $k\in \mathbb{N}$, monotonically converges to the solution of (4). Hence, by the inverse change of variable (see again [11,14] for details)

$$u=\frac{ln\left(\varphi \right)}{{\sigma }^2},m=\varphi \psi ,$$

(7)

we obtain a solution of the original problem (3). We have

Theorem 1.

The sequence $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$ defined by (5) and (6) converges in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and a.e. to a weak solution $(\varphi ,\psi )$ of (4). Moreover, the couple $(u,m)$ defined by (7) is a weak solution to (3).

The main difficulty in the study of problems (3) and (4) is due both in the degeneracy of the second order operator with respect to x and in the unbounded dependence of the coefficients of the first order terms with respect to v. To overcome the previous difficulties we rely on the results for linear kinetic Fokker–Planck equations developed in [18]. We mention that existence of weak solutions for the standard MFG problem, possibly degenerate, has been studied in [19], but the results in this paper do not cover the present setting. The previous iterative procedure also suggests a monotone numerical method for the approximation of (4), hence for (3). Indeed, by approximating (5) and (6) by finite differences and solving alternatively the resulting discrete equations, we obtain an approximation of the sequence $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$. A corresponding procedure for the standard quadratic MFG system was studied in [14], where the convergence of the method is proved. We plan to study the properties of the previous numerical procedure in a future work.

2. Well Posedness of the Kinetic Fokker–Planck System

In this section, we study the existence of a solution to system (4). The proof of the result follows the strategy implemented in Section 2 of [14] for the case of a standard MFG system with quadratic Hamiltonian and relies on the results for linear kinetic Fokker–Planck equations in Appendix A of [18]. We remark the model here studied does not fit exactly the problem treated in [18] because of the presence of a zero order term in the Fokker-Planck equation. Hence some technical aspects should be analyzed in more detail, however the present paper is mainly intended to give some idea on the change of variabile for the kinetic MGF.

We fix the assumptions we will assume in the whole paper. The vector field $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and the coupling cost $f:{\mathbb{R}}^d\times {\mathbb{R}}^d\times \mathbb{R}\to \mathbb{R}$ are assumed to satisfy

$$\begin{array}{cc}& b\in L^{\infty }\left({\mathbb{R}}^d\right),\\ & f\in L^{\infty }({\mathbb{R}}^d\times {\mathbb{R}}^d\times \mathbb{R}),f\le 0\mathrm{and}f(x,v,·)\mathrm{strictly}\mathrm{decreasing}.\end{array}$$

Moreover, the diffusion coefficient $\sigma$ is positive and the initial and terminal data satisfy

$$\begin{array}{c}\hfill m_0\in L^{\infty }({\mathbb{R}}^d\times {\mathbb{R}}^d),m_0\ge 0,\int \int m_0(x,v)dxdv=1,\\ \hfill \mathrm{and}\exists R_0>0\mathrm{s}.\mathrm{t}.\mathrm{supp}\left\{m_0\right\}\subset {\mathbb{R}}^d\times B(0,R_0)\end{array}$$

(8)

and

$$\begin{array}{c}\hfill u_T\in C^0({\mathbb{R}}^d\times {\mathbb{R}}^d)\mathrm{and}\exists C_0,C_1>0\mathrm{s}.\mathrm{t}.\forall (x,v)\in {\mathbb{R}}^d\times {\mathbb{R}}^d\\ \hfill -C_0{\left(\right|v|}^2+\left|x\right|)-C_0\le u_T(x,v)\le -C_1{\left(\right|v|}^2+\left|x\right|)+C_1.\end{array}$$

(9)

Note that (9) implies that $e^{u_T/{\sigma }^2}\in L^{\infty }({\mathbb{R}}^d\times {\mathbb{R}}^d)\cap L^2({\mathbb{R}}^d\times {\mathbb{R}}^d)$. We denote with $(·,·)$ the scalar product in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and with $\langle ·,·\rangle$ the pairing between $\mathcal{X}=L^2([0,T]\times {\mathbb{R}}_x^d;H^1\left({\mathbb{R}}_v^d\right))$ and its dual ${\mathcal{X}}^{\prime }=L^2([0,T]\times {\mathbb{R}}_x^d;H^{-1}\left({\mathbb{R}}_v^d\right))$. We define the following functional space

$$\mathcal{Y}=\left\{g\in L^2([0,T]\times {\mathbb{R}}_x^d,H^1\left({\mathbb{R}}_v^d\right)),{\partial }_{t}g+v·D_{x}g\in L^2([0,T]\times {\mathbb{R}}_x^d,H^{-1}\left({\mathbb{R}}_v^d\right))\right\}$$

and we set ${\mathcal{Y}}_0=\{g\in \mathcal{Y}:g\ge 0\}$. If $g\in \mathcal{Y}$, then it admits (continuous) trace values $g(0,x,v)$, $g(T,x,v)\in L^2({\mathbb{R}}^d\times {\mathbb{R}}^d)$ (see [18], Lemma A.1) and therefore the initial/terminal conditions for (4) are well defined in $L^2$ sense. We first prove the well posedness of problems (5) and (6).

Proposition 2.

We have

(i) 

For any $\psi \in {\mathcal{Y}}_0$, there exists a unique solution $\varphi \in {\mathcal{Y}}_0$ to

$$\left\{\begin{array}{c}{\partial }_t\varphi +\frac{{\sigma }^2}{2}{\Delta }_v\varphi -b\left(x\right)·D_v\varphi +v·D_x\varphi =-\frac{1}{{\sigma }^2}f(x,v,\psi \varphi )\varphi \hfill \\ \varphi (T,x,v)=e^{\frac{u_T(x,v)}{{\sigma }^2}}.\hfill \end{array}\right.$$

(10)

Moreover, $\varphi \in L^{\infty }([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and, for any $R>0$, there exist ${\delta }_R\in \mathbb{R}$ and $\rho >0$ such that

$$\varphi (t,x,v)\ge C_R:=e^{\frac{1}{{\sigma }^2}({\delta }_R-\rho T)}\forall t\in [0,T],(x,v)\in B(0,R)\subset {\mathbb{R}}^d\times {\mathbb{R}}^d.$$

(11)

(ii) 

Let $\mathsf{\Phi }:{\mathcal{Y}}_0\to {\mathcal{Y}}_0$ be the map which associates to ψ the unique solution of (10). Then, if ${\psi }_2\le {\psi }_1$, we have $\mathsf{\Phi }\left({\psi }_2\right)\ge \mathsf{\Phi }\left({\psi }_1\right)$.

Proof. 

We first prove existence of a solution to the nonlinear problem (10) by a fixed point argument exploiting the results for the corresponding linear problem proved in [18]. Fixed $\psi \in {\mathcal{Y}}_0$, consider the map $F=F\left(\phi \right)$ from $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ into itself that associates with $\phi$ the weak solution $\varphi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ of the linear problem

$$\left\{\begin{array}{c}{\partial }_t\varphi +\frac{{\sigma }^2}{2}{\Delta }_v\varphi -b\left(x\right)·D_v\varphi +v·D_x\varphi =-\frac{1}{{\sigma }^2}f\left(\psi \phi \right)\varphi \hfill \\ \varphi (T,x,v)=e^{\frac{u_T(x,v)}{{\sigma }^2}}.\hfill \end{array}\right.$$

(12)

By Prop. A.2 of [18], $\varphi$ belongs to $\mathcal{Y}$ and it coincides with the unique solution of (12) in this space. Moreover, the following estimate

$${\parallel \varphi \parallel }_{L^2([0,T]\times {\mathbb{R}}_x^d;H^1\left({\mathbb{R}}_v^d\right))}+{\parallel {\partial }_t\varphi +v·D_x\varphi \parallel }_{L^2([0,T]\times {\mathbb{R}}_x^d;H^{-1}\left({\mathbb{R}}_v^d\right))}\le C$$

(13)

holds for some constant C which depends only on $\parallel e^{u_T/{\sigma }^2}{\parallel }_{L^2}$, ${\parallel f\parallel }_{L^{\infty }}$ and $\sigma$. Hence F maps $B_C$, the closed ball of radius C of $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, into itself.

To show that the map F is continuous on $B_C$, consider ${\left\{{\phi }_n\right\}}_{n\in \mathbb{N}},\phi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ such that $\parallel {\phi }_n{- \phi \parallel }_{L^2}\to 0$ and set ${\varphi }_n=F\left({\phi }_n\right)$. Then ${\varphi }_n\in \mathcal{Y}$, and, by the estimate (13), we get that, up to a subsequence, there exists $\overline{\varphi }\in \mathcal{Y}$ such that ${\varphi }_n\to \overline{\varphi }$, $D_v{\varphi }_n\to D_v\overline{\varphi }$ in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, ${\partial }_t{\varphi }_n+v·D_x{\varphi }_n\to {\partial }_t{\overline{\varphi }}_n+v·D_x{\overline{\varphi }}_n$ in $L^2([0,T]\times {\mathbb{R}}_x^d;H^{-1}\left({\mathbb{R}}_v^d\right))$. Moreover, ${\phi }_n\to \phi$ almost everywhere. By the definition of weak solution to (12), we have that

$$\langle {\partial }_t{\varphi }_n+v·D_x{\varphi }_n,w\rangle -\frac{{\sigma }^2}{2}(D_v{\varphi }_n,D_{v}w)-(b·D_v{\varphi }_n,w)=(-\frac{1}{{\sigma }^2}{\varphi }_{n}f\left({\phi }_n\psi \right),w),$$

(14)

for any $w\in \mathcal{D}([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, the space of infinite differentiable functions with compact support in $[0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d$. Employing weak convergence for left hand side of (14) and the Dominated Convergence Theorem for the right hand one, we get for $n\to \infty$

$$\langle {\partial }_t\overline{\varphi }+v·D_x\overline{\varphi },w\rangle -\frac{{\sigma }^2}{2}(D_v\overline{\varphi },D_{v}w)-(b·D_v\overline{\varphi },w)=(-\overline{\varphi }f\left(\phi \psi \right),w)$$

for any $w\in \mathcal{D}([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$. Hence $\overline{\varphi }=F\left(\phi \right)$ and $F\left({\phi }_n\right)\to F\left(\phi \right)$ for $n\to \infty$ in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$. The compactness of the map F in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ follows by the compactness of the set of the solutions to (12), see Theorem 1.2 of [20]. We conclude, by Schauder’s Theorem, that there exists a fixed-point of the map F in $L^2$, hence in $\mathcal{Y}$, and therefore a solution to the nonlinear parabolic Equation (10).

Observe that, if $\varphi$ is a solution of (10), then $\tilde{\varphi }=e^{\lambda t}\varphi$ is a solution of

$${\partial }_t\tilde{\varphi }+\frac{{\sigma }^2}{2}{\Delta }_v\tilde{\varphi }-b\left(x\right)·D_v\tilde{\varphi }+v·D_x\tilde{\varphi }-\lambda \tilde{\varphi }=-\frac{1}{{\sigma }^2}f(e^{-\lambda t}\psi \tilde{\varphi })\tilde{\varphi }$$

(15)

with the corresponding final condition. In the following, we assume that $\lambda >0$. To show that $\varphi$ is non-negative, we will exploit the following property (see Lemma A.3 of [18]): given $\varphi \in \mathcal{Y}$ and defined ${\varphi }^{\pm }=max(\pm \varphi ,0)$, then ${\varphi }^{\pm }\in \mathcal{X}$ and

$$\langle {\partial }_t\varphi +v·D_x\varphi ,{\varphi }^{-}\rangle =\frac{1}{2}\left({\int \int |\varphi (0,x,v)}^{-}{|}^{2}dxdv-\int \int {\left|\varphi {(T,x,v)}^{-}\right|}^{2}dxdv\right).$$

(16)

Let $\varphi$ be a solution of (15), multiply the equation by ${\varphi }^{-}$ and integrate. Then, since $\varphi (T,x,v)$ is non-negative, by (16) we get

$$\begin{array}{c}\hfill -\frac{1}{{\sigma }^2}(\varphi f\left(e^{\lambda t}\varphi \psi \right),{\varphi }^{-})=\langle {\partial }_t\varphi +v·D_x\varphi ,{\varphi }^{-}\rangle -\\ \hfill \frac{{\sigma }^2}{2}(D_v\varphi ,D_v{\varphi }^{-})-(b·D_v\varphi ,{\varphi }^{-})-\lambda (\varphi ,{\varphi }^{-})=\\ \hfill \frac{1}{2}\int \int {\left|\varphi {(0,x,v)}^{-}\right|}^{2}dxdv+\frac{{\sigma }^2}{2}(D_v{\varphi }^{-},D_v{\varphi }^{-})+\lambda ({\varphi }^{-},{\varphi }^{-})\ge \\ \hfill \lambda ({\varphi }^{-},{\varphi }^{-}),\end{array}$$

where it has been exploited that, by integration by parts, $(b·D_v\varphi ,{\varphi }^{-})=0$. Since $f\le 0$ and therefore

$$-(\varphi f\left(e^{\lambda t}\varphi \psi \right),{\varphi }^{-})=({\varphi }^{-}f\left(e^{\lambda t}\varphi \psi \right),{\varphi }^{-})\le 0,$$

we get $({\varphi }^{-},{\varphi }^{-})\equiv 0$, hence $\varphi \ge 0$.

To prove the uniqueness of the solution to (10), consider two solutions ${\varphi }_1$, ${\varphi }_2$ of (15) and set $\overline{\varphi }={\varphi }_1-{\varphi }_2$. Multiplying the equation for $\overline{\varphi }$ by $\overline{\varphi }$, integrating and using $\overline{\varphi }(x,v,T)=0$, we get

$$\begin{array}{c}\hfill -\frac{1}{{\sigma }^2}(f\left(e^{-\lambda t}\psi {\varphi }_1\right){\varphi }_1-f\left(e^{-\lambda t}\psi {\varphi }_2\right){\varphi }_2,{\varphi }_1-{\varphi }_2)=\langle {\partial }_t\overline{\varphi }+v·D_x\overline{\varphi },\overline{\varphi }\rangle -\\ \hfill \frac{{\sigma }^2}{2}(D_v\overline{\varphi },D_v\overline{\varphi })-(b·D_v\overline{\varphi },\overline{\varphi })-\lambda (\overline{\varphi },\overline{\varphi })=\\ \hfill -\frac{1}{2}\int \int {\left|\overline{\varphi }(x,v,0)\right|}^{2}dxdv-\frac{{\sigma }^2}{2}(D_v\overline{\varphi },D_v\overline{\varphi })-\lambda (\overline{\varphi },\overline{\varphi })\le -\lambda ({\varphi }_1-{\varphi }_2,{\varphi }_1-{\varphi }_2)\end{array}$$

(17)

and, by the strict monotonicity of f, we conclude that ${\varphi }_1={\varphi }_2$.

To prove that $\varphi$ is bounded from above, we observe that the function $\overline{\varphi }(t,x,v)=e^{C_1+(T-t){\parallel f\parallel }_{\infty }/{\sigma }^2}$, where $C_1$ as in (9), is a supersolution of the linear problem (12) for any $\phi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, i.e., $\varphi (T,x,v)\ge e^{u_T(x,v)/{\sigma }^2}$ and

$${\partial }_t\overline{\varphi }+\frac{{\sigma }^2}{2}{\Delta }_v\overline{\varphi }-b\left(x\right)·D_v\overline{\varphi }+v·D_x\overline{\varphi }\le -\frac{1}{{\sigma }^2}f\left(\psi \phi \right)\overline{\varphi }.$$

By the Maximum Principle (see Prop. A.3 (i) in [18]), we get that $\overline{\varphi }\ge \varphi$, where $\varphi$ is the solution of (12). Since the previous property holds for any $\phi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, we conclude that $\overline{\varphi }\ge \varphi$, where $\varphi$ is the solution of the nonlinear problem (10).

A similar argument show that $\underline{\varphi }(x,v,t)=e^{(-C_0{\left(\right|v|}^2+\left|x\right|+1)-\rho (T-t))/{\sigma }^2}$, where $C_0$ as in (9) and $\rho$ sufficiently large, is a subsolution of (12) for any $\phi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$. Indeed, replacing $\underline{\varphi }$ in the equation, we get that the inequality

$$\begin{array}{c}\hfill {\partial }_t\underline{\varphi }+\frac{{\sigma }^2}{2}{\Delta }_v\underline{\varphi }-b\left(x\right)·D_v\underline{\varphi }+v·D_x\underline{\varphi }=\\ \hfill =\frac{\underline{\varphi }}{{\sigma }^2}\left(\rho -C_{0}d{\sigma }^2+2C_0^2{\sigma }^2{\left|v\right|}^2+2C_{0}b\left(x\right)·v-C_{0}v·\frac{x}{\left|x\right|}\right)\ge \\ \hfill -\frac{1}{{\sigma }^2}f\left(\psi \phi \right)\underline{\varphi }\end{array}$$

is satisfied for $\rho$ large enough and, moreover, $\underline{\varphi }\le e^{u_T(x,v)/{\sigma }^2}$. Hence $\underline{\varphi }\le \varphi$, where $\varphi$ is the solution of the nonlinear problem (10), and, from this estimate, we deduce (11).

We finally prove the monotonicity of the map $\mathsf{\Phi }$. Set ${\varphi }_i=\mathsf{\Phi }\left({\psi }_i\right)$, $i=1,2$, and consider the equation satisfied by $\overline{\varphi }=e^{\lambda t}{\varphi }_1-e^{\lambda t}{\varphi }_2$, multiply it by ${\overline{\varphi }}^{+}$ and integrate. Performing a computation similar to (17), we get

$$\begin{array}{c}\hfill -\frac{1}{{\sigma }^2}(f\left({\varphi }_1{\psi }_1\right){\varphi }_1-f\left({\varphi }_2{\psi }_2\right){\varphi }_2,{\overline{\varphi }}^{+})\le -\lambda ({\overline{\varphi }}^{+},{\overline{\varphi }}^{+}).\end{array}$$

Since, by monotonicity of f and non-negativity of ${\varphi }_i$, we have

$$\begin{array}{c}\hfill -(f\left({\varphi }_1{\psi }_1\right){\varphi }_1-f\left({\varphi }_2{\psi }_2\right){\varphi }_2,{\overline{\varphi }}^{+})=-(f\left({\varphi }_1{\psi }_1\right)({\varphi }_1-{\varphi }_2),{\overline{\varphi }}^{+})-\\ \hfill ((f\left({\varphi }_1{\psi }_1\right)-f\left({\varphi }_2{\psi }_2\right)){\varphi }_2,{\overline{\varphi }}^{+})\ge 0,\end{array}$$

we get $({\overline{\varphi }}^{+},{\overline{\varphi }}^{+})=0$ and therefore ${\varphi }_1\le {\varphi }_2$. □

We set

$${\mathcal{Y}}_R=\{\varphi \in {\mathcal{Y}}_0:\varphi \ge C_R\forall (x,v)\in B(0,R),t\in [0,T]\},$$

where $C_R$ is defined as in (11).

Proposition 3.

Given $R>R_0$, where $R_0$ as in (8), we have

(i) 

For any $\varphi \in {\mathcal{Y}}_R$, there exists a unique solution $\psi \in {\mathcal{Y}}_0$ to

$$\left\{\begin{array}{c}{\partial }_t\psi -\frac{{\sigma }^2}{2}{\Delta }_v\psi -b\left(x\right)·D_v\psi +v·D_x\psi =\frac{1}{{\sigma }^2}f(x,v,\psi \varphi )\psi \hfill \\ \psi (0,x,v)=\frac{m_0(x,v)}{\varphi (0,x,v)}.\hfill \end{array}\right.$$

(18)

Moreover

$$\psi (x,v,t)\le \frac{\parallel m_0{\parallel }_{L^{\infty }}}{C_R}\forall t\in [0,T],(x,v)\in {\mathbb{R}}^d\times {\mathbb{R}}^d,$$

(19)

where $C_R$ as in (11).

(ii) 

Let $\mathsf{\Psi }:{\mathcal{Y}}_R\to {\mathcal{Y}}_0$ be the map which associates with $\varphi \in {\mathcal{Y}}_R$ the unique solution of (18). Then, if ${\varphi }_2\le {\varphi }_1$, we have $\mathsf{\Psi }\left({\varphi }_2\right)\ge \mathsf{\Psi }\left({\varphi }_1\right)$.

Proof. 

First observe that, since $R>R_0$, then $\psi (0,x,v)$ is well defined for $\varphi \in {\mathcal{Y}}_R$. The proof of the first part of $\left(i\right)$ is very similar to the one of the corresponding result in Proposition 2, hence we only prove the bound (19). If $\psi$ is a solution of (18), then $\tilde{\psi }=e^{-\lambda t}\psi$ is a solution of

$${\partial }_t\tilde{\psi }-\frac{{\sigma }^2}{2}{\Delta }_v\tilde{\psi }-b\left(x\right)·D_v\tilde{\psi }+v·D_x\psi +\lambda \tilde{\psi }=\frac{1}{{\sigma }^2}f(x,v,e^{\lambda t}\tilde{\psi }\varphi )\psi .$$

(20)

Let $\psi$ be a solution of (20), set $\overline{\psi }=\psi -e^{-\lambda t}{\parallel m_0\parallel }_{L^{\infty }}/C_R$ and observe that $\overline{\psi }\left(0\right)\le 0$. Multiply the equation for $\overline{\psi }$ by ${\overline{\psi }}^{+}$ and integrate to obtain

$$\begin{array}{c}\hfill (\psi f\left(e^{\lambda t}\psi \varphi \right),{\overline{\psi }}^{+})=\\ \hfill \langle {\partial }_t\overline{\psi }+v·D_x\overline{\psi },{\overline{\psi }}^{+}\rangle +\frac{1}{{\sigma }^2}(D_v\overline{\psi },D_v{\overline{\psi }}^{+})-(b\left(x\right)D_v\overline{\psi },{\overline{\psi }}^{+})+\lambda (\overline{\psi },{\overline{\psi }}^{+})\ge \\ \hfill \int \int |{\overline{\psi }}^{+}{(x,v,T)|}^{2}dxdv+\lambda ({\overline{\psi }}^{+},{\overline{\psi }}^{+})\ge \lambda ({\overline{\psi }}^{+},{\overline{\psi }}^{+}).\end{array}$$

Since $\psi \ge 0$ and $f\le 0$, we have

$$(\psi f\left(e^{\lambda t}\psi \varphi \right),{\overline{\psi }}^{+})\le 0$$

and therefore ${\overline{\psi }}^{+}\equiv 0$. Hence the upper bound (19).

Now we prove (ii). Set ${\psi }_i=\mathsf{\Psi }\left({\varphi }_i\right)$, $i=1,2,$ and $\overline{\psi }=e^{-\lambda t}{\psi }_1-e^{-\lambda t}{\psi }_2$. Multiply the equation satisfied by $\overline{\psi }$ by ${\overline{\psi }}^{+}$ and integrate. Since, by monotonicity and negativity of f, we have

$$\begin{array}{c}\hfill (f\left(e^{\lambda t}{\varphi }_1{\psi }_1\right){\psi }_1-f\left(e^{\lambda t}{\varphi }_2{\psi }_2\right){\psi }_2,{\overline{\psi }}^{+})=(f\left(e^{\lambda t}{\varphi }_1{\psi }_1\right)({\psi }_1-{\psi }_2),{\overline{\psi }}^{+})+\\ \hfill ({\psi }_2(f\left(e^{-\lambda t}{\varphi }_1{\psi }_1\right)-f\left(e^{-\lambda t}{\varphi }_2{\psi }_2\right)),{\overline{\psi }}^{+})\le 0.\end{array}$$

Then

$$\begin{array}{c}\hfill 0\ge \langle {\partial }_t\overline{\psi }+v·D_x\overline{\psi },{\overline{\psi }}^{+}\rangle +\frac{1}{{\sigma }^2}(D_v\overline{\psi },D_v{\overline{\psi }}^{+})-(b\left(x\right)D_v\overline{\psi },{\overline{\psi }}^{+})+\lambda (\overline{\psi },{\overline{\psi }}^{+})\ge \\ \hfill \int \int |{\overline{\psi }}^{+}{(x,v,T)|}^{2}dxdv+\lambda ({\overline{\psi }}^{+},{\overline{\psi }}^{+})\ge \lambda ({\overline{\psi }}^{+},{\overline{\psi }}^{+}).\end{array}$$

Hence ${\overline{\psi }}^{+}\equiv 0$ and therefore ${\psi }_1\le {\psi }_2$. □

Proof of Theorem 1. 

Given ${\psi }^{\left(0\right)}\equiv 0$, consider the sequence $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$, $k\in \mathbb{N}$, defined in (5) and (6). It can rewritten as

$$\left\{\begin{array}{c}{\varphi }^{(k+\frac{1}{2})}=\mathsf{\Phi }\left({\psi }^{\left(k\right)}\right)\hfill \\ {\psi }^{(k+1)}=\mathsf{\Psi }({\varphi }^{(k+\frac{1}{2})})\hfill \end{array}\right.$$

(21)

where the maps $\mathsf{\Phi }$, $\mathsf{\Psi }$ are as in Propositions 2 and, respectively 3. Observe that, by (11), we have ${\varphi }^{(k+\frac{1}{2})}\in {\mathcal{Y}}_R$ for $R>R_0$ and ${\psi }^{(k+1)}\ge 0$ for any k. Hence the sequence $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$ is well defined. We first prove by induction the monotonicity of the components of $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$. By non-negativity of solutions to (18), we have ${\psi }^{\left(1\right)}=\mathsf{\Phi }({\varphi }^{(\frac{1}{2})})\ge 0$ and therefore ${\psi }^{\left(1\right)}\ge {\psi }^{\left(0\right)}$. Moreover, by the monotonicity of $\mathsf{\Phi }$, ${\varphi }^{\left(\frac{3}{2}\right)}=\mathsf{\Phi }\left({\psi }^{\left(1\right)}\right)\le \mathsf{\Phi }\left({\psi }^{\left(0\right)}\right)={\varphi }^{\left(\frac{1}{2}\right)}$. Now assume that ${\psi }^{(k+1)}\ge {\psi }^{\left(k\right)}$. Then

$${\varphi }^{(k+\frac{3}{2})}=\mathsf{\Phi }({\psi }^{(k+1)})\le \mathsf{\Phi }({\psi }^{\left(k\right)})={\varphi }^{(k+\frac{1}{2})}$$

and

$${\psi }^{(k+2)}=\mathsf{\Psi }({\varphi }^{(k+\frac{3}{2})})\ge \mathsf{\Psi }({\varphi }^{(k+\frac{1}{2})})={\psi }^{(k+1)},$$

therefore the monotonicity of two sequences.

Since ${\varphi }^{(k+\frac{1}{2})}\ge 0$ and, by (19), for $k\to \infty$, the sequence ${\psi }^{(k+1)}\le {\parallel m_0\parallel }_{L^{\infty }}/C_R$, $({\varphi }^{(k+\frac{1}{2})},{\psi }^{(k+1)})$ converges a.e. and in $L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ to a couple $(\varphi ,\psi )$. Taking into account the estimate (13), the a.e. convergence of the two sequences and repeating an argument similar to the one employed for the continuity of the map F in Proposition 2, we get that the couple $(\varphi ,\psi )$ satisfies, in weak sense, the first two equations in (4). The terminal condition for $\varphi$ is obviously satisfied, while the initial condition for $\psi$, in $L^2$ sense, follows by convergence of ${\varphi }^{(k+\frac{1}{2})}\left(0\right)$ to $\varphi \left(0\right)$.

We now consider the couple $(u,m)$ given by the change of variable in (7). We first observe that, by Theorem 1.5 of [10], we have ${\partial }_t\varphi +v·D_x\varphi$, $D_v\varphi$, ${\Delta }_v\varphi \in L^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and a corresponding regularity for $\psi$. Taking into account the boundedness of $\varphi$ and the estimate in (11), we have that u, ${\partial }_{t}u+v·D_{x}u$, $D_{v}u$, ${\Delta }_{v}u\in L_{loc}^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$. Hence we can write the equation for u in weak form, i.e.,

$$({\partial }_{t}u+v·D_{x}u,w)-\frac{{\sigma }^2}{2}(D_{v}u,D_{v}w)-(b·D_{v}u,w)+\frac{1}{2}\left(\right|D_{v}u{|}^2,w)=-(f\left(m\right),w),$$

for any $w\in \mathcal{D}([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$, with final datum in trace sense. In a similar way, since m, ${\partial }_{t}m+v·D_{x}m$, $D_{v}m$, ${\Delta }_{v}m\in L_{loc}^2([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ and m is locally bounded, we can rewrite also the equation for m in weak form, i.e.,

$$({\partial }_{t}m+v·D_{x}m,w)+\frac{{\sigma }^2}{2}(D_{v}m,D_{v}w)-(b·D_{v}m,w)-(m{D}_{v}u,Dw)=0,$$

for any $w\in \mathcal{D}([0,T]\times {\mathbb{R}}^d\times {\mathbb{R}}^d)$ with the initial datum in trace sense. □