Ergodic theorems for transient one-dimensional diffusions
type: Item
instance of: scholarly article
title: Ergodic theorems for transient one-dimensional diffusions
zbMATH Open document ID: 0835.60070
DOI: 10.1016/0304-4149(94)00084-7
author: K. B. Athreya
author: Ananda Weerasinghe
published in: Stochastic Processes and their Applications
publication date: 1996-03-26T00:00:00Z
review text: For a one-dimensional diffusion \(X = \{X(t), t \geq 0\}\) satisfying \[ dX(t) = \mu \bigl( X(t) \bigr)dt+ \sigma \bigl( X(t) \bigr) dB(t), \;t \geq 0,\quad X(0) = x_0, \] \(B\) a Brownian motion on the line, the following problem is considered: Under which conditions on \(\mu\), \(\sigma\) and a Borel set \(B\) a relation of the form \[ \int^t_0 1_B \bigl( X(u) \bigr) du \sim \text{const.} t \;(t \to \infty)\;P_x \text{-a.s.} \] holds? The conditions on \(\mu\) and \(\sigma\) are not explicitly formulated in the context of Feller's or Dynkin's treatment of diffusions but here for short the terminology of them is used. Basically it is assumed that, with respect to the scale \(S\) and the speed measure \(m\) of \(X\), \(- \infty\) is an entrance or natural boundary and \(+ \infty\) is a natural boundary with \(S(+ \infty) < + \infty\) (implying transience of \(X\), infinite lifetime of \(X\) and \(X(t) \to + \infty\), \(P_x\)-a.s.). Then some further conditions are given relating the scale, the speed measure and the set \(B\) near \(+ \infty\) in such an appropriate manner that ergodic relations of the wanted kind hold.
reviewed by: Wilfried Schenk
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 803384
zbMATH Keywords: local time
zbMATH Keywords: occupation time
zbMATH Keywords: Brownian motion
zbMATH Keywords: diffusions
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00084-7
OpenAlex ID: W2046029856
cites work: Probabilistic and Deterministic Averaging
cites work: Q4011272
cites work: Q3911791
cites work: Q4039796
Recommended article: Ergodicity of transition semigroups for stochastic fast diffusion equations
Recommended article: On quasi-ergodic distribution for one-dimensional diffusions
Recommended article: Ergodicity of one-dimensional regime-switching diffusion processes
Recommended article: Ergodic theorems for reaction-diffusion processes
Recommended article: A note on the quasi-ergodic distribution of one-dimensional diffusions
Recommended article: Ergodic theorems for reaction diffusion processes
Recommended article: Ergodicity of Finite-Energy Diffusions
Recommended article: Q4842803
Recommended article: Limit theorems for transient diffusions on the line
description: scientific article; zbMATH DE number 803384