Hitting probabilities for Brownian motion on Riemannian manifolds.
type: Item
instance of: scholarly article
title: Hitting probabilities for Brownian motion on Riemannian manifolds.
zbMATH Open document ID: 1042.58022
DOI: 10.1016/S0021-7824(01)01244-2
author: A. A. Grigor'yan
author: Laurent Saloff-Coste
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-25T00:00:00Z
review text: Let \(M\) be a complete non-compact Riemannian manifold, with heat kernel \(p(t,x,y)\) and Wiener measure \(\mathbb{P}_x\), and \(K\) be a compact subset of \(M\). The aim of this article is to estimate the hitting probability function: \(\psi(t,x):= \mathbb{P}_x((\exists s\in [0,t]) X_s\in K)\), when \(t> 0\) and \(d(x,K)\geq \delta> 0\), and also \(\partial_t\psi(t,x)\). As a main tool, the authors first establish the following potential-theoretic formula: for any precompact neighborhood of \(K\), say open \(L\), and for any \(x\in M\setminus K\), \(t> 0\): \[ \partial_t \psi(t,x)= 2\int_{L\setminus K} p_K(t,x,\cdot)_x |\nabla\varphi|^2 \,d\mu- \int_{L\setminus K}\partial_t p_K(t,x,\cdot)_x \varphi^2 \,d\mu, \] where \(p_K(t,x,y)\) is the Dirichlet heat kernel of \(L\), relative to some fixed measure \(\mu\) on \(M\), and where \(\varphi\) is the equilibrium potential of the capacitor \((K,L)\). Many fine estimates are then deduced, in various cases and ranges, for \(\psi\) and \(\partial_t\psi\), from both sides. In particular it is proved that if \(\text{cap}(K,L)> 0\) and \(x\not\in L\): \[ \psi(t,x)\leq 2\text{\,cap}(K,L)\times \int^t_0 \sup_{y\in L\setminus K}\, p_K(s,x,y)\,ds \] and \[ \psi(\infty,x)- \psi(t,x)\leq 2\text{\,cap}(K, L)\times \int^\infty_t \sup_{y\in L\setminus K}\, p_K(s,x,y)\,ds+ \mu(L\setminus K)\sup_{y\in L\setminus K}\, p_K(t,x,y). \] In the transient case, it is shown that \[ \psi(t,x)\geq \text{cap}(K)\times \int^t_0 \inf_{y\in\partial K} p(s,x,y)\,ds \] and \[ \psi(\infty, x)- \psi(t,x)\leq \text{cap}(K)\times \int^\infty_t \sup_{y\in\partial K}\, p(s,x,y)\,ds. \] More specific bounds are obtained under geometric conditions on \(M\), in both recurrent and transient cases. In the example \(M= \mathbb{R}^2\), \(K= \langle| x|\leq 1\}\), the following bounds are obtained: \(c_0\exp(- C_1| x|^2/t)\leq \log| x|\times \psi(t, x)\leq C_1\exp(- C_0| x|^2/t)\) if \(0< t< 2| x|^2\); and \(\psi(t, x)\approx (\log\sqrt{t}- \log| x|)/\log\sqrt{t}\)…
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 31C12
zbMATH DE Number: 1985988
zbMATH Keywords: Brownian motion
zbMATH Keywords: hitting probabilities
zbMATH Keywords: heat kernel
zbMATH Keywords: potential theory
reviewed by: Jacques Franchi
MaRDI profile type: Publication
Wikidata QID: Q115340563
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full work available at URL: s0021-7824(01)01244-2
OpenAlex ID: W2038181363
description: scientific article; zbMATH DE number 1985988