Let $\mathcal P_2$ be the space of probability measures $\mu$ on $\mathbb
R^d$ with $\mu(|\cdot|^2)<\infty$. Consider the following time-dependent second
order differential operator on $\mathbb R^d\times\mathcal P_2:$ $${\bf L}_t f
(x,\mu):= \frac 1 2\big\<\bar a(t,x,\mu), \nabla^2 f(x,\mu)\big\>_{HS} +
\big\<\bar b(t,x,\mu), \nabla f(x,\mu)\big\> $$ $$\qquad + \int_{\mathbb R^d}
\Big[\ff 1 2 \big\
_{HS}
+\big\\Big]\mu(\d y),$$ where $t\ge 0, $
$\nabla$ is the gradient operator in $x\in\mathbb R^d$, $D$ is the intrinsic
derivative in $\mu\in\mathcal P_2$, introduced by Albeverio, Kondratiev and the
second named author in 1996. Furthermore, $$b,\bar b: [0,\infty)\times \mathbb
R^d\times\mathcal P_2\to \mathbb R^d,\ \ a,\bar a: [0,\infty)\times \mathbb
R^d\times\mathcal P_2\to \mathbb R^{d}\otimes\mathbb R^d$$ are measurable with
$a$ and $\bar a$ non-negative definite. We investigate the existence,
uniqueness and exponential ergodicity of the diffusion process generated by
${\bf L}_t$, and use the diffusion process to solve the following PDE on
$[0,T]\times \mathbb R^d\times\mathcal P_2$: $$(\partial_t+{\bf L}_t)u + V u
+f=0,$$ where $V$ and $f$ are functions on $[0,T]\times \mathbb
R^d\times\mathcal P_2.$