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

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 $\frac12Δ+b(x)\nabla$ 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 $B^-(r)=\min_{|x|=r}b(x)\cdot\frac x{|x|}$.

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