Asymptotic capacities for finely divided bodies and stopped diffusions
type: Item
instance of: scholarly article
title: Asymptotic capacities for finely divided bodies and stopped diffusions
zbMATH Open document ID: 0618.60072
author: J. R. Baxter
author: Naresh C. Jain
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: In \({\mathbb{R}}^ d\), \(d\geq 3\), let \((B_ t)\) denote the Brownian motion and let \((D_ n)\) be a sequence of closed sets. Let \((\tau_ n)\) denote the first hitting-time of \(D_ n\). The authors introduced the notion of ''stable convergence'' of \((\tau_ n)\) to a randomized stopping-time T. Consider the following boundary values problem (BVP) for the heat equation: (1)\ \(\partial V(x,t)/\partial t=(1/2)\Delta v(x,t)\) for \(t>0\), \(x\in D_ n^ c\), \(v(x,t)=0\) for \(t>0\), \(x\in \partial D_ n\), \(v(x,0)=f(x)\) for \(x\in D^ c_ n\), where \(f: {\mathbb{R}}^ d\to {\mathbb{R}}\) is a bounded Borel function. It is known that the above BVP admits \(u_ n(x,t)=E^ x(f(B_ t)\chi_{(\tau_ n>t)})\) as solution. The authors show that if \((\tau_ n)\) converges stably to T then, \[ u(x,t)=E^ x(f(B_ t)\chi_{(T>t)}) \] is the solution of the following BVP: \[ \partial v(x,t)/\partial t=(1/2)\Delta v(x,t)- h(x)v(x,t)\quad for\quad t>0,\quad x\in {\mathbb{R}}^ d,\quad v(x,0)=f(x)\quad for\quad x\in {\mathbb{R}}^ d, \] where the killing rate h represents a ''limiting capacity density''. The model could be used to study certain random media problems which consists of many small bodies becoming smaller and more densely distributed as \(n\to \infty\).
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4003197
zbMATH Keywords: first hitting-time
zbMATH Keywords: randomized stopping-time
zbMATH Keywords: boundary values problem
zbMATH Keywords: killing rate
zbMATH Keywords: random media problems
reviewed by: Xuan Loc Nguyen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4003197