Transience/recurrence and growth rates for diffusion processes in time-dependent regions

Abstract

Let $\mathcal{K} \subset R^d$, $d\ge 2$, be a smooth, bounded domain satisfying $0\in \mathcal{K} $, and let $f(t),\ t\ge 0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)\mathcal{K} \subset R^d$. Consider a diffusion process corresponding to the generator $\frac 12\Delta +b(x)\nabla $ in the time-dependent region $\bar D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac 12\frac{d^2} {dx^2}+B(x)\frac d{dx}$ on the time-dependent region $[1,f(t)]$ with reflection at the boundary. We give precise conditions for transience/recurrence of the one-dimensional process in terms of the growth rates of $B(x)$ and $f(t)$. In the recurrent case, we also investigate positive recurrence, and in the transient case, we also consider the asymptotic growth rate of the process. Using the one-dimensional results, we give conditions for transience/recurrence of the multi-dimensional process in terms of the growth rates of $B^+(r)$, $B^-(r)$ and $f(t)$, where $B^+(r)=\max _{|x|=r}b(x)\cdot \frac x{|x|}$ and $B^-(r)=\min _{|x|=r}b(x)\cdot \frac x{|x|}$.