Asymptotic properties of some multidimensional diffusions
type: Item
instance of: scholarly article
title: Asymptotic properties of some multidimensional diffusions
zbMATH Open document ID: 0622.60062
author: Charles R. Clark
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: Let \(X_ t\in {\mathbb{R}}^ d\) be the solution to the stochastic differential equation \[ dX_ t=\sigma (X_ t)dB_ t+b(X_ t)dt,\quad X_ 0\in {\mathbb{R}}^ d, \] where \(B_ t\) is a Brownian motion in \({\mathbb{R}}^ d\). The aim of the paper is to make the following statement precise: ''Let \(x_ t\) be a solution of \(\dot x=b(x)\). If \(| x_ t| \to \infty\) as \(t\to \infty\) and the drift vector field b(x) is well behaved near \(x_ t\) then with positive probability, \(X_ t\to \infty\), and does so asymptotically like \(x_ t.''\) Examples are provided to illustrate the situations in which this theorem may be applied.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 4009459
zbMATH Keywords: transience
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: stochastic differential equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992076
OpenAlex ID: W2061204791
DOI: 10.1214/AOP/1176992076
description: scientific article; zbMATH DE number 4009459