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

DOI10.1214/16-EJP4355zbMATH Open1345.60091arXiv1505.04764MaRDI QIDQ303563

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Publication date: 22 August 2016

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Abstract: Let $mathcalKsubset{R}^d$, $dge2$, be a smooth, bounded domain satisfying $0inmathcalK$, and let $f(t),tge0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)mathcalKsubset{R}^d$. Consider a diffusion process corresponding to the generator $frac12Delta+b(x)abla$ 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 $frac12frac{d}^{2}d{x}^2+B(x)fracddx$ 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 $B^{+}(r)=ma{x}_{|x|=r}b(x)cdotfracx|x|$ and $B^{-}(r)=mi{n}_{|x|=r}b(x)cdotfracx|x|$.

Full work available at URL: https://arxiv.org/abs/1505.04764

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