Transience/Recurrence and Growth Rates for Diffusion Processes in Time-Dependent Domains

TL;DR

This paper establishes criteria for the transience and recurrence of diffusion processes in time-dependent domains, analyzing how growth rates of domain size and drift influence long-term behavior.

Contribution

It provides new precise conditions for transience and recurrence of diffusion processes in evolving domains, extending understanding to multi-dimensional cases.

Findings

Conditions for transience depend on growth rates of domain and drift.

Criteria for recurrence and positive recurrence are established.

Asymptotic growth rates of the process are characterized in transient cases.

Abstract

Let $\mathcal{K}\subset R^d$, $d\ge2$, be a smooth, bounded domain satisfying $0\in\mathcal{K}$, and let $f(t),\ t\ge0$, 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 $\frac12\Delta+b(x)\nabla$ in the time-dependent domain $D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac12\frac{d^2}{dx^2}+B(x)\frac d{dx}$ on the time-dependent domain $(1,f(t))$ 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 $B^-(r)=\min_{|x|=r}b(x)\cdot\frac x{|x|}$.