m}{\partial t}+\mathcal{A}^{*}m+\text{div }(G(x,m,Du)m(x)) =0,\\
m(x,0)=m_{0}(x);
\end{array}
\right.$$

form the system of coupled HJB- FP equations of the classical mean field type control problem, see [@BFY].

\[sub:MEAN-FIELD-GAMES\]MEAN FIELD GAMES 
-----------------------------------------

In Mean Field Games, we cannot have a Bellman equation, similar to (\[eq:2.6\]), (\[eq:2.7-1\]), since the problem is not simply a control problem. However, if we first fixed parameter $m(\cdot)$ in (\[eq:1.3\]), (\[eq:1.4\]) we have a standard control problem. We introduce the state dynamics and the cost functional accordingly $$\left\{
\begin{split}
dx(s)&=g(x(s),m,v(x(s)))ds+\sigma(x(s))dw,\label{eq:1.3-1}\\
x(t) &=x;
\end{split}
\right.$$ $$J_{x,t}(v(\cdot),m(\cdot))= \mathbb{E} \bigg[\int_{t}^{T}f(x(s),m(s),v(x(s)))ds+h(x(T),m(T))\bigg].\label{eq:1.4-1}$$

If we set $$u(x,t)=\inf_{v(\cdot)}J_{x,t}(v(\cdot),m(\cdot))$$ in which we omit to write explicitly the dependence of $u$ in $m.$ Then $u(x,t)$ satisfies Bellman equation $$\left\{
\begin{split}
-\dfrac{\partial u}{\partial t}+\mathcal{A}u&=H(x,m,Du(x)),\label{eq:2.12-1}\\
u(x,T)&=h(x,m).
\end{split}
\right.$$

For mean field games, we next require that $m$ must be the probability density of the optimal state, hence $$\left\{
\begin{array}{l}
\dfrac{\partial m}{\partial t}+\mathcal{A}^{*}m+\text{div }(G(x,m,Du)m(x)) =0,\label{eq:2.14-1}\\
m(x,0)=m_{0}(x);
\end{array}
\right.$$ and this is the system of HJB-FP equations, corresponding to the classical Mean Field Games problem. We can check that, if one considers the Master equation $$\label{eq:2.10-1}
\left\{
\begin{array}{l}
-\dfrac{\partial U}{\partial t}+\mathcal{A}U+\displaystyle\int_{\mathbb{R}^{n}}\frac{\partial}{\partial m}U(x,m,t)(\xi)(\mathcal{A}^{*}m(\xi)+\text{div (}G(\xi,m,DU(\xi))m(\xi))d\xi\\
\qquad=H(x,m,DU(x)),\\
U(x,m,T)=h(x,m);
\end{array}
\right.$$ then $$u(x,t)=U(x,m(t),t)\label{eq:2.12-2}.$$ Combining (\[eq:2.10-1\]) and (\[eq:2.14-1\]), we obtain easily (\[eq:2.12-1\]), and hence (\[eq:2.12-2\]) if the 