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Ergodicity and Feyman-Kac Formula for Space-Distribution Valued Diffusion Processes

Research paper by Panpan Ren, Michael Rockner, Feng-Yu Wang

Indexed on: 16 Apr '19Published on: 14 Apr '19Published in: arXiv - Mathematics - Probability



Abstract

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 $\mathbb R^d\times\mathcal P_2:$ $${\bf L}_t f (x,\mu):= \frac 1 2\big\<\bar a(t,x,\mu), \nabla^2 f(x,\mu)\big\>_{HS} + \big\<\bar b(t,x,\mu), \nabla f(x,\mu)\big\> $$ $$\qquad + \int_{\mathbb R^d} \Big[\ff 1 2 \big\<a(t,y,\mu), \nabla \{(D f (x,\mu))(\cdot)\}(y)\big\>_{HS} +\big\<b(t,y,\mu), (D f(x,\mu))(y)\big\>\Big]\mu(\d y),$$ 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 $[0,T]\times \mathbb R^d\times\mathcal P_2.$