m(\xi)+\text{div}(G(\xi,m,DU(\xi))m(\xi))\Big)d\xi\\
    \qquad\qquad+\beta^{2}\text{div}\:(\displaystyle\int_{\mathbb{R}^{n}}\frac{\partial U(x,m,t)}{\partial m}(\xi)Dm(\xi)d\xi)-\frac{1}{2}\beta^{2}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{\partial^{2}U(x,m,t)}{\partial m^{2}}(\xi,\eta)Dm(\xi)Dm(\eta)d\xi d\eta\\
    \qquad=H(x,m,DU(x)),\\
    U(x,m,T)=h(x,m).\\  
\end{array}
\right.$$

\[Rem1\]It is very important to notice that the property $$\frac{\partial U(x,m,t)}{\partial m}(\xi)=\frac{\partial U(\xi,m,t)}{\partial m}(x)\label{eq:4.8}$$ is true for the Master equation of the Mean Field type control problem (\[eq:2.8-1\]) and not true for the Master equation of the Mean Field games problem, (\[eq:4.7\]). We shall notice this discrepancy in the case of linear quadratic (LQ) problems, in which explicit formulas can be obtained.

REMARKS ON THE APPROACH OF CARMONA-DELARUE [@CAD]
-------------------------------------------------

Independently of our work, Carmona and Delarue [@CAD] have put recently on Arxiv, an article in which they try to interpret the Master equation differently. To synthesize our approach, we state that there is a Bellman equation for the Mean field type control problem, in which the space variable is in an infinite dimensional space. The Master equation is obtained by taking the gradient in this space variable. This differentiation is taken in the sense of Frechet. It is also possible to derive from the Master equation a coupled system of stochastic HJB-FP equations, in which the solution of the HJB equation appears not as a value function, but as an adjoint equation for an infinite dimentional stochastic control problem. 