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2016 Transience/recurrence and growth rates for diffusion processes in time-dependent regions
Ross G. Pinsky
Electron. J. Probab. 21: 1-24 (2016). DOI: 10.1214/16-EJP4355

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 $B^-(r)=\min _{|x|=r}b(x)\cdot \frac x{|x|}$.

Citation

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Ross G. Pinsky. "Transience/recurrence and growth rates for diffusion processes in time-dependent regions." Electron. J. Probab. 21 1 - 24, 2016. https://doi.org/10.1214/16-EJP4355

Information

Received: 11 June 2015; Accepted: 22 June 2016; Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 1345.60091
MathSciNet: MR3530323
Digital Object Identifier: 10.1214/16-EJP4355

Subjects:
Primary: 60J60

JOURNAL ARTICLE
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Vol.21 • 2016
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