Quenched critical large deviations for Brownian motion in a Poissonian potential
type: Item
instance of: scholarly article
title: Quenched critical large deviations for Brownian motion in a Poissonian potential
zbMATH Open document ID: 0853.60027
published in: Journal of Functional Analysis
publication date: 1996-03-20T00:00:00Z
review text: The author considers a \(d\)-dimensional Brownian motion \(\{Z_t \}_{t \geq 0}\) moving in a random potential \(V\). The potential \(V(x, \omega)\equiv (W* \omega) (x)\), \(x\in \mathbb{R}^d\), consists of a bounded, nonnegative, compactly supported shape function \(W\) convoluted with the cloud configuration \(\omega= \sum_i \delta_{x_i}\) of a Poisson point process of constant intensity. For every \(t\geq 0\) and configuration \(\omega\), a weighted path measure \(Q_{t, \omega}\) is defined by introducing a weight proportional to \(\exp(- \int^t_0 V(Z_s, \omega)ds)\) for every Brownian path. Building on his previous work, the author proves for almost all cloud configurations \(\omega\), that the measures \(\{Q_{t, \omega} Y_t^{-1} \}_{t> 0}\) satisfy a large deviation principle on \(\mathbb{R}^d\) with scale \(t(\log t)^{-2/ d}\) as \(t\to \infty\), where \(Y_t \equiv (1/t) (\log t)^{2/d} Z_t\) is the correspondingly scaled position of the Brownian motion at time \(t\). This result is applied to determine the off-diagonal asymptotic behaviour of the Schrödinger heat kernel and the quenched large deviations of Brownian motion with drift in the above random potential \(V\).
reviewed by: Uwe Schmock
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 847458
zbMATH Keywords: random environment
zbMATH Keywords: quenched behaviour
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: weighted path measures
zbMATH Keywords: soft obstacles
zbMATH Keywords: Schrödinger heat kernel
zbMATH Keywords: Brownian motion with drift
author: Alain-Sol Sznitman
MaRDI profile type: Publication
full work available at URL: jfan.1995.1082
OpenAlex ID: W2077450027
DOI: 10.1006/JFAN.1995.1082
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description: scientific article; zbMATH DE number 847458