Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type
type: Item
instance of: scholarly article
title: Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type
zbMATH Open document ID: 0911.60065
author: Toshiro Watanabe
published in: Journal of the Mathematical Society of Japan
publication date: 1999-05-16T00:00:00Z
review text: A stochastic process of Ornstein-Uhlenbeck type \((X_t)^\infty_0\) on a \(d\)-dimensional space is considered. It can be represented as \[ X_t=e^{-tQ}x+\int^t_0e^{-(t-u)Q}dZ_u, \] where \(Q\) is a real \(d\times d\) matrix all eigenvalues of which have positive real parts, \((Z_t)^\infty_0\) is a Lévy process on \(R^d\), i.e. a stochastically continuous process with stationary independent increments, and \(Z_0=0\). It is proved that \((X_t)\) is recurrent if and only if \((\forall_c>0)\) \[ \int^1_0{dv\over v}\exp\left[\int^1_v{du\over u}\int_{| x|\geq c}\biggl(\exp\bigl(-| e^{(\log u)Q}x|\bigr)-1\biggr)\rho(dx)\right]=\infty, \] where \(\rho\) is a measure of Lévy of the process \((Z_t)\). Some sufficient conditions for this equality to be fulfilled are derived.
reviewed by: B. P. Harlamov
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 1185920
zbMATH Keywords: stochastic integral
zbMATH Keywords: Lévy process
Wikidata QID: Q123143025
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05010155
OpenAlex ID: W2066407111
DOI: 10.2969/JMSJ/05010155
Recommended article: Recurrence conditions for multidimensional processes of Ornstein- Uhlenbeck type
Recommended article: Q4024302
Recommended article: Q4320189
Recommended article: Q4284313
Recommended article: On the interval recurrence property of \((N,d)\)-Ornstein-Uhlenbeck processes
Recommended article: Q4240223
Recommended article: On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes
Recommended article: A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type
Recommended article: Q4021341
description: scientific article; zbMATH DE number 1185920