N,$ $$\left\{
\begin{array}{l}
-\displaystyle\frac{\partial u^{i}}{\partial t}-\frac{1}{2}\sum_{j=1}^{N}\text{tr}(a(x^{j})D_{x^{j}}^{2}u^{i})-\frac{\beta^{2}}{2}\text{tr}\sum_{j,k=1}^{N}D_{x^{j}x^{k}}^{2}u^{i}-\sum_{j\not=i}D_{x^{j}}u^{i}.G(x,D_{x^{j}}u^{j})=H(x,D_{x^{i}}u^{i}),\label{eq:6.4}\\
u^{i}(x,T)=h(x),
\end{array}
\right.$$ and define the feedbacks $$\hat{v}^{i}(x)=\hat{v}(x,D_{x^{i}}u^{i}(x)),\label{eq:6.5}$$ which form a Nash equilibrium for the differential game (\[eq:6.1\]), (\[eq:6.2\]). This means first that

$$u^{i}(x,t)=J^{i}(x,t;\hat{v}(\cdot)),\label{eq:6.6}$$

and if we use the notation $$v(x)=(v^{i}(x),\bar{v}^{i}(x)),$$ in which $\bar{v}^{i}(x)$ represents all the components of $v(x)$, except $v^{i}(x)$, to emphasize the control of player $i$, then $$u^{i}(x,t)\leq J^{i}(x,t;v^{i}(x),\bar{\hat{v}}^{i}(x)),\:\forall v^{i}(x)\label{eq:6.7}.$$

We next apply this framework in the following case. Consider $f(x,m,v)$, $g(x,m,v),h(x,m)$ as in the previous sections (with $x\in \mathbb{R}^{n})$, and assume this time that the argument $m$ is no more an element of $L^{2}(\mathbb{R}^{n})$ but a probability measure in $\mathbb{R}^{n}$. We define $$\begin{aligned}
f(x,v^{i}(x)) & =f(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},v^{i}(x)),\quad \, g(x,v^{i}(x))=g(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},v^{i}(x)),\\
h(x) & =h(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}}),\end{aligned}$$ in which $x=(x^{1},\cdots,x^{N}).$ The Hamiltonian becomes $$H(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},q)=\inf_{v}(f(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},v)+q.g(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},v)).$$

Consider the optimal feedback $\hat{v}(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},q)$, we have $$G(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},q)=g(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},\hat{v}(x^{i},\frac{1}{N-1}\sum_{j\not=i}\delta_{x^{j}},q)).$$

Hence the system of PDEs as in (\[eq:6.4\]) becomes, for $i=1, \cdots, N,$ $$\label{eq:6.8}
\left\{
\begin{array}{l}
-\displaystyle\frac{\partial u^{i}}{\partial t}-\frac{1}{2}\sum_{j=1}^{N}\text{tr}(a(x^{j})D_{x^{j}}^{2}u^{i})-\frac{\beta^{2}}{2}\text{tr}\sum_{j,k=1}^{N}D_{x^{j}x^{k}}^{2}u^{i}-\sum_{j\not=i}D_{x^{j}}u^{i}.G(x^{i},\frac{1}{N-1}\sum_{k\not=j}\delta_{x^{k}},D_{x^{j}}u^{j})\\
\qquad=H(x^{i},\frac{1}{N-1}\displaystyle\sum_{j\not=i}\delta_{x^{j}},D_{x^{i}}u^{i}), \\
u^{i}(x,T)=h(x,\frac{1}{N-1}\displaystyle\sum_{j\not=i}\delta_{x^{j}}).\\ 
\end{array}
\right.$$

We want to interpret (\[eq:6.8\]) as a Master equation, which is the way P.L. Lions has introduced the concept of Master equation.

\[sub:DISCUSSION-ON-THE\]DISCUSSION ON THE DERIVATIVE WITH RESPECT TO A PROBABILITY MEASURE. 
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For a functional $F(m)$ where $m$ is in $L^{2}(\mathbb{R}^{n})$, we have defined the concept of derivative by 