Small perturbation of stochastic parabolic equations: A power series analysis
type: Item
instance of: scholarly article
title: Small perturbation of stochastic parabolic equations: A power series analysis
zbMATH Open document ID: 1009.60052
published in: Journal of Functional Analysis
publication date: 2002-11-28T00:00:00Z
review text: The author investigates a parameter-depending SPDE of the form \[ \begin{aligned} du^\varepsilon(t,x)=&\left(\sum_{i,j}a^{ij,\varepsilon} (t,x)D_iD_ju^\varepsilon +f^\varepsilon(t,x,Du^\varepsilon,u^\varepsilon)\right) dt\\ &+\sum_k\left(\sum_i\sigma^{ik,\varepsilon}(t,x)D_iu^\varepsilon+ g^{k,\varepsilon}(t,x,u^\varepsilon)\right) dw_k(t),\end{aligned} \] for \(t>0\), \(x\in {\mathbb R}^d\), and \(u^\varepsilon(0,x)=\varphi^\varepsilon(x)\). Here, \(w_k\) are independent standard Wiener processes (their number may be infinite), \(D_i=\partial/\partial x_i\), \(i,j=1,\dots,d\), and all coefficients can depend on \(\omega\). The main result is that, under certain conditions, \(u^\varepsilon=\sum_{m=0}^n\varepsilon^mu^{(m)}+ o(\varepsilon^n)\), where \(u^{(m)}\) are defined recursively as solutions of linear equations.
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1836674
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: small perturbation
zbMATH Keywords: \(L_p\)-estimate
author: S. V. Lototsky
reviewed by: Tomasz Bojdecki
MaRDI profile type: Publication
OpenAlex ID: W1997809532
cites work: Stochastic flows and Taylor series
cites work: Large deviations for a Burgers'-type SPDE
cites work: Uniform large deviations for parabolic SPDEs and applications
cites work: Q3134548
cites work: Q4692653
cites work: Q4399897
cites work: Q3672830
cites work: Q4086525
cites work: Singularly perturbed diffusisons: Rapid switchings and fast diffusions.
cites work: Large deviations for a class of stochastic partial differential equations
cites work: Asymptotic Series for Singularly Perturbed Kolmogorov–Fokker–Planck Equations
cites work: On Transition Densities of Singularly Perturbed Diffusions with Fast and Slow Components
cites work: Asymptotic Expansions of Singularly Perturbed Systems Involving Rapidly Fluctuating Markov Chains
cites work: Q3875932
cites work: Large deviations for a reaction-diffusion equation with non-Gaussian perturbations
cites work: An introduction to the theory of large deviations
cites work: Q3341599
cites work: Asymptotic probabilities and differential equations
cites work: Singularly perturbed multidimensional switching diffusions with fast and slow switchings
DOI: 10.1006/JFAN.2001.3923
description: scientific article; zbMATH DE number 1836674