Asymptotic series and exit time probabilities
type: Item
instance of: scholarly article
title: Asymptotic series and exit time probabilities
zbMATH Open document ID: 0771.60055
author: Matthew R. James
author: Wendell Fleming
published in: The Annals of Probability
publication date: 1993-01-17T00:00:00Z
review text: The authors consider a bounded domain \(D\subset R^ n\), a stochastic equation \[ dx^ \varepsilon_ t=b^ \varepsilon(x_ t,t)dt+\varepsilon^{1/2}\sigma (x^ \varepsilon_ t)dw_ t, \] \(x_ s^ \varepsilon=x\), \(x\in D\) etc., for \(\varepsilon\to 0\), \(\tau^ \varepsilon=\inf\{t;x_ t\notin D\}\) and \(q^ \varepsilon(x,s)=P^ \varepsilon_{x,s}(\tau^ \varepsilon\leq T)\), where \(T\) is fixed and \(P^ \varepsilon_{x,s}\) is the distribution of the solution. They suppose that \(a(x)=\sigma(x)\sigma'(x)\geq\alpha_ 0\cdot 1\) for all \(x,\alpha_ 0>0\), that \(\sigma\) is \(C^ \infty\) and bounded, that \(b^ \varepsilon\) is continuous in \((s,x)\), Lipschitz in \(x\) uniformly in \(\varepsilon,s\), that \(b\cdot\nu<0\) where \(\nu\) is the exterior normal on \(\partial D\). Near 0, \(b^ \varepsilon\) is expressed as \(b+\varepsilon^{1/2}b_ 1+\cdots+\varepsilon^{m/2}b_ m+o(\varepsilon^{m/2})\) uniformly, with \(b_ i\) in \(C^ \infty\). The result is \[ q^ \varepsilon(x,s)=\exp(-u(x,s)\varepsilon^{-1}- v(x,s)\varepsilon^{-1/2}-w(x, s))(1+\varepsilon^{1/2}\varphi_ 1(x,s)+\cdots+\varepsilon^{m/2}\varphi_ m(x,s)+o( \varepsilon^{m/2})),\tag{1} \] where \(u\) appears from the application of a large deviations result in [\textit{M. I. Frejdlin} and \textit{A. D. Venttsel'}, Random perturbations of dynamical systems (1984; Zbl 0522.60055)], while \(v,w,\varphi_ i\) are solutions of partial derivative equations in which the \(s\)-derivatives appear only as \((\partial/\partial s)+\cdots\). The proof uses the fact that \(q^ \varepsilon\) satisfies the equation \(\partial q/\partial s+(\varepsilon/2)tr(a(x)D^ 2q)+b^ \varepsilon\cdot Dq=0\), \(q=1\) on \(\partial Dx[0,T)\), \(q(x,T)=0\) for \(x\in D\). The result is preceded by two sections, concerning solutions of such equations, on \(R^ n\) instead of \(D\). In the first one a result of the type \(q^ \varepsilon\to q\) is established by probabilistic methods, in the second one a result analogous to (1) but without \(\varepsilon\) under e…
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 97398
zbMATH Keywords: diffusion with small parameter
zbMATH Keywords: large deviation principle
zbMATH Keywords: random perturbations
zbMATH Keywords: region of strong regularity
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989695
OpenAlex ID: W2022386023
DOI: 10.1214/AOP/1176989695
description: scientific article; zbMATH DE number 97398