I do not understand the following when reading a paper on Propagation of Chaos, written by A.S.Sznitman:
Consider an $n$- dimensional process $X$ satisfying the following SDE: $$ dX_t = b(t, X_t, \mathbb{P}_t) \,dt + \sigma (t, X_t , \mathbb{P}_t) \,dW_t, \quad \quad \mathbb{P}_t = \text{Law} \, (X_t), \quad X_0 \in \mathbb{R}^n,$$ where $W$ is a $d$-dimensional Brownian motion, $b^i (t ,x, \mathbb{P}_t)= \int b^i (t,x,y) \mathbb{P}_t (dy)$ and ${\sigma}^i_j (t ,x, \mathbb{P}_t)= \int {\sigma}^i_j (t,x,y) \mathbb{P}_t (dy)$.
Suppose that $X=(X_t)_{t \in [0,T]}$ satisfies the above SDE. If for some measure $\mathbb{Q}$ on $C([0,T], \mathbb{R}^n)$, the following SDE (interpreted similarly as above) is satisfied: $$ dX_t = b(t, X_t , \mathbb{Q}_t) \,dt + \sigma ( t, X_t , \mathbb{Q}_t ) \,dW_t,$$ then the paper claims that $\mathbb{Q}= \text{Law} (X_t: 0 \leq t \leq T)$.
I have no idea how to show this rigorously. Any suggestions?
Hi this kinds of SDE are called McKean-Vlasov SDE, you can have a look here and there and to the references therein.
Best regards.