On the spatial asymptotic behavior of stochastic flows in Euclidean space
type: Item
instance of: scholarly article
title: On the spatial asymptotic behavior of stochastic flows in Euclidean space
zbMATH Open document ID: 0942.60053
author: P. Imkeller
author: Michael Scheutzow
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: The authors study asymptotic growth rates of stochastic flows on \(\mathbb R^d\) and their derivatives with respect to the spatial parameter under Lipschitz conditions on the local characteristics of the generating semimartingales. It is asked: Can we obtain exact rates of growth by applying the Garsia-Rodemich-Rumsey lemma (GRR lemma) in its majorizing measure version to moment inequalities for the flow, while carefully keeping track of the quality of the constants appearing in the main martingale inequalities of Doob and Burkholder, Davis and Gundy? The main task is to show that the answer to this question is ``yes''. The majorizing measure form of the GRR lemma is used to derive moduli of continuity for random fields on \(\mathbb R^d\) with values in some metric space \((M,\rho)\) which satisfy moment conditions of the form \[ E[\rho(\varphi(x), \varphi(y))^p]\leq d(x, y)^p \exp(cp^2), \] either for some fixed \(p\) and the metric \(d(x, y) = |x - y|\) or \(\forall p\geq 1\) and the metric \(d(x, y) = |x - y|\) or \(\forall p\geq 1\) and the metric \(d(x, y) = |x - y|\wedge 1\), \(x, y \in M.\) The moment conditions in the second case are verified for the stochastic flow \(\varphi\) generated by a stochastic differential equation driven by a semimartingale \(F\) whose local characteristics satisfy suitable Lipschitz conditions. Together with a similar estimate on the inverse of the modulus of the flow it is obtained a theorem, which shows that the supremum of \(|\varphi_{st}(x)|\) over \(0\leq s, t\leq T\) is bounded by a random variable \(Y\) multiplied by \(|x|\exp(\gamma(\ln\ln|x|)^{1/2})\) as \(|x|\to\infty\) for some constant \(\gamma > 0,\) and moreover that \(Y\) is integrable with respect to a certain Young function growing faster than any polynomial. The corresponding questions on the growth rate of higher derivatives of the flow \(\varphi\) are considered too. Under differentiability conditions on the local characteristics of \(F,\) the growth rate of a…
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1359642
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: stochastic flow
zbMATH Keywords: spatial asymptotic growth rate
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Garsia-Rodemich-Rumsey lemma
zbMATH Keywords: random fields
zbMATH Keywords: semimartingale
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677255
OpenAlex ID: W2016106491
cites work: Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory
cites work: On the integrability condition in the multiplicative ergodic theorem for stochastic differential equations
cites work: Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times
cites work: Occupation densities of stratonovitch stochastic differential equations with boundary conditions
cites work: Q3995203
cites work: Q3997990
cites work: The lyapunov spectrum and stable manifolds for stochastic linear delay equations
cites work: Q4010326
cites work: Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. I: The multiplicative ergodic theory
cites work: Spatial estimates for stochastic flows in Euclidean space
cites work: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
DOI: 10.1214/AOP/1022677255
Recommended article: Spatial estimates for stochastic flows in Euclidean space
Recommended article: Q3742390
Recommended article: Q5748020
Recommended article: On stochastic Euler equation in \(\mathbb R^d\)
Recommended article: Q4010321
Recommended article: Asymptotic expansion of stochastic flows
Recommended article: Ergodicity with respect to the spatial variable of discrete-time stochastic flows
description: scientific article; zbMATH DE number 1359642