its adjoint

$$\mathcal{A}^{*}\varphi(x)=-\frac{1}{2}\sum_{i,j}\frac{\partial^{2}(a_{ij}(x)\varphi(x))}{\partial x_{i}\partial x_{j}}.$$

The mean field type control problem is to minimize the cost functional $$\label{eq:1.2}
J(v(\cdot))= \mathbb{E} \bigg[\int_{0}^{T}f(x(t),m_{v(\cdot)}(t),v(x(t)))dt+h(x(T),m_{v(\cdot)}(T))\bigg].$$

In the classical mean field games problem, one should first fix $m(\cdot)\in C([0,T];L^{2}(\mathbb{R}^{n}))$ as a given parameter in the state equation $$\label{eq:1.3}
\left\{
\begin{array}{rcl}
    dx&=&g(x,m,v(x))dt+\sigma(x)dw,\\
    x(0) & =&x_{0},\\ 
\end{array}
\right.$$ and the objective functional $$\label{eq:1.4}
J(v(\cdot),m(\cdot))= \mathbb{E} \bigg[\int_{0}^{T}f(x(t),m(t),v(x(t)))dt+h(x(T),m(T))\bigg].$$

The mean field games problem looks for an equilibrium $\hat{v}(\cdot),m(\cdot)$ such that $$\left\{
\begin{array}{l}
J(\hat{v}(\cdot),m(\cdot)) \leq J(v(\cdot),m(\cdot)),\:\forall v(\cdot),\\
 m(t)  \text{ is the probability density of }\hat{x}(t),\,\forall t\in[0,T],\label{eq:1.5}
\end{array}
\right.$$ where $\hat{x}(\cdot)$ is the solution of (\[eq:1.3\]) corresponding to the equilibrium pair $\hat{v}(\cdot),m(\cdot)$.

MASTER EQUATION FOR THE CLASSICAL CASE $ $
==========================================

We refer to problems (\[eq:1.1\]),(\[eq:1.2\]) and (\[eq:1.3\]),(\[eq:1.4\]),(\[eq:1.5\]) as the classical case. We define the Hamiltonian $H(x,m,q):\, \mathbb{R}^{n}\times L^{2}(\mathbb{R}^{n})\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ as $$H(x,m,q)=\inf_{v}(f(x,m,v)+q\cdot g(x,m,v))$$ and the optimal value of $v$ is denoted by $\hat{v}(x,m,q).$ We then set $$G(x,m,q)=g(x,m,\hat{v}(x,m,q)).$$

MEAN FIELD TYPE CONTROL PROBLEM
-------------------------------

The mean field type control problem is easily transformed into a control problem in which the state is a probability density process $m_{v(\cdot)}$, which satisfies the solution of the Fokker-Planck equation $$\left\{
\begin{split}
&\frac{\partial m_{v(\cdot)}}{\partial t}+\mathcal{A}^{*}m_{v(\cdot)}+\text{div }(g(x,m_{v(\cdot)},v(x))m_{v(\cdot)}(x))=0\label{eq:2.1},\\
&m_{v(\cdot)}(x,0)=m_{0}(x). 
\end{split}
\right.$$

Here $m_{0}(x)$ is the density of the 