Small perturbation of diffusions in inhomogeneous media
type: Item
instance of: scholarly article
title: Small perturbation of diffusions in inhomogeneous media
zbMATH Open document ID: 1002.60070
DOI: 10.1016/S0246-0203(01)01101-3
author: Tzuu-Shuh Chiang
author: S. J. Sheu
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2002-09-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77717
review text: Let \(dX^\varepsilon (t)=b(X^\varepsilon (t))dt+\varepsilon \sigma(X^\varepsilon(t))dW(t)\), \(t\in[0,1]\), \(X^\varepsilon(0)=x^0\in R^d,\) be a system of \(d\)-dimensional stochastic differential equations, where \(b(x)\) and \(\sigma (x)\) are smooth except possibly along the hyperplane \(\{(x_1,\dots,x_d); x_1=0\}\). The authors will demonstrate that the natural setup of its large deviation principle is to consider the probability \(\varepsilon^2\log P(\|X^\varepsilon-\varphi\|<\delta, \|u^\varepsilon-\psi\|<\delta, \|l^\varepsilon-\eta\|<\delta)\sim-I(\varphi,\psi,\eta)\) of the triplet \((X^\varepsilon, u^\varepsilon, l^\varepsilon)\) simultaneously. Here, \(u^\varepsilon\) is the occupation time of \(X^\varepsilon_1(\cdot)\) in the positive half line and \(l^\varepsilon(\cdot)\) is the local time of \(X^\varepsilon_1(\cdot)\) at 0. The explicit form of the rate function \(I(\cdot,\cdot,\cdot)\) is obtained. The usual Wentzell-Freidlin theory concerns only probabilities of the form \(\varepsilon^2\log P(\|X^\varepsilon-\varphi\|<\delta)\) and its limit is a consequence of the contraction principle of the authors' result.
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 1751662
zbMATH Keywords: Wentzell-Freidlin theory
zbMATH Keywords: large deviation principle
reviewed by: N. G. Gamkrelidze
MaRDI profile type: Publication
OpenAlex ID: W2072452459
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description: scientific article; zbMATH DE number 1751662