$m$ to obtain the master equation. We note that $$\begin{array}{rcl}
&&\dfrac{\partial}{\partial m}\left(\displaystyle\int_{\mathbb{R}^{n}}\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(\xi,m,t)}{\partial m}(\eta)Dm(\xi)Dm(\eta)d\xi d\eta\right)(x)\\
&=&\displaystyle\int_{\mathbb{R}^{n}}\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial^{2}U(x,m,t)}{\partial m^{2}}(\xi,\eta)Dm(\xi)Dm(\eta)d\xi d\eta-2\text{div}\left(\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(x,m,t)}{\partial m}(\eta)Dm(\eta)d\eta\right),
\end{array}$$ and $$\begin{array}{rcl}
&&\dfrac{\partial}{\partial m}\left(\displaystyle\int_{\mathbb{R}^{n}}H(\xi,m,DU)m(\xi)dx\right)(x)\\
&=&H(x,m,DU(x))+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}H(\xi,m,DU(\xi))(x)m(\xi)d\xi\\
&&\qquad +\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}(DU(\xi,m,t))(x)G(\xi,m,DU(\xi))m(\xi)d\xi.
\end{array}$$

Next, using (\[add\_3\]) in terms of those in the present setting, we have $$\int_{\mathbb{R}^{n}}\frac{\partial}{\partial m}(DU(\xi,m,t))(x)G(\xi,m,DU(\xi))m(\xi)d\xi=\int_{\mathbb{R}^{n}}D_{\xi}(\frac{\partial}{\partial m}U(\xi,m,t)(x))G(\xi,m,DU(\xi))m(\xi)d\xi$$ $$=-\int_{\mathbb{R}^{n}}\frac{\partial}{\partial m}U(\xi,m,t)(x)\:\text{div}(G(\xi,m,DU(\xi))m(\xi))d\xi=-\int_{\mathbb{R}^{n}}\frac{\partial}{\partial m}U(x,m,t)(\xi)\:\text{div}(G(\xi,m,DU(\xi))m(\xi))d\xi.$$

Collecting results, we obtain the Master equation $$\label{eq:2.8-1}
\left\{
\begin{array}{l}
    -\dfrac{\partial U}{\partial t}+\mathcal{A}U-\dfrac{1}{2}\beta^{2}\Delta U\\
    \qquad+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}U(x,m,t)(\xi)\left(\mathcal{A}^{*}m(\xi)-\dfrac{1}{2}\beta^{2}\Delta m(\xi)+\text{div (}G(\xi,m,DU(\xi))m(\xi))\right)d\xi\\
    \qquad-\dfrac{1}{2}\beta^{2}\displaystyle\int_{\mathbb{R}^{n}}\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial^{2}U(x,m,t)}{\partial m^{2}}(\xi,\eta)Dm(\xi)Dm(\eta)d\xi d\eta+\beta^{2}\text{div}\left(\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(x,m,t)}{\partial m}(\xi)Dm(\xi)d\xi\right)\\
    =H(x,m,DU(x))+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}H(\xi,m,DU(\xi))(x)m(\xi)d\xi;\\
    U(x,m,T)=h(x,m)+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial h(\xi,m)}{\partial m}(x)m(\xi)d\xi.
\end{array}
\right.$$

We note that this equation reduces to (\[eq:2.10\]) when $\beta=0$.

SYSTEM OF HJB-FP EQUATIONS 
---------------------------

We first check that we can derive from the Master equation a system of coupled stochastic HJB-FP equations. Consider the conditional probability density process corresponding to the optimal feedback $\hat{v}(x,m,DU(x,m,t))$, which is the solution of $$\left\{
\begin{array}{l}
\partial_{t}m+(\mathcal{A}^{*}m-\dfrac{1}{2}\beta^{2}\Delta m+\text{div}(G(x,m,DU)m))dt+\beta Dm.db(t)=0\label{eq:3.6}\\
m(x,0)=m_{0}(x).\\
\end{array}
\right.$$

Set $u(x,t)=U(x,m(t),t)$, we obtain $$\label{eq:3.7}
\left\{
\begin{array}{l}
    -\partial_{t}u+(\mathcal{A}u-\dfrac{1}{2}\beta^{2}\Delta u)dt +\beta^{2}\text{div}\left(\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(x,m,t)}{\partial m}(\xi)Dm(\xi)d\xi\right)dt\\
    \qquad=\bigg(H(x,m,Du(x))+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}H(\xi,m,Du(\xi))(x)m(\xi)d\xi\bigg)dt+\beta\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial U(x,m,t)}{\partial m}(\xi)Dm(\xi)d\xi db(t)\\
    u(x,T)=h(x,m)+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial h(\xi,m)}{\partial m}(x)m(\xi)d\xi.
\end{array}
\right.$$

Let $$B(x,t)=\int_{\mathbb{R}^{n}}\frac{\partial U(x,m,t)}{\partial m}(\xi)Dm(\xi)d\xi,$$ we can rewrite (\[eq:3.6\]), (\[eq:3.7\]) as follows, by noting the Itô’s correction term of $u$ that involves the second derivative of $U$ with respect to $m$, $$\label{eq:3.7-1}
\begin{array}{l}
\left\{
\begin{array}{l}
-\partial_{t}u+(\mathcal{A}u-\dfrac{1}{2}\beta^{2}\Delta u)dt +\beta^{2}\text{div}B(x,t)dt\\
\qquad=\bigg(H(x,m,Du(x))+\displaystyle\int_{\mathbb{R}^{n}}\dfrac{\partial}{\partial m}H(\xi,m,Du(\xi))(x)m(\xi)d\xi\bigg)dt+\beta B(x,t) 