Generalized solutions of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.35065
DOI: 10.1142/S0219493712003523
author: Yana Belopolskaya
author: Wojbor A. Woyczyński
published in: Stochastics and Dynamics
publication date: 2012-06-04T00:00:00Z
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35K45
zbMATH DE Number: 6042627
zbMATH Keywords: stochastic flows
zbMATH Keywords: backward Kolmogorov equation
zbMATH Keywords: two-component Markov process
zbMATH Keywords: Kunita theory
MaRDI profile type: Publication
cites work: Generalized solutions of nonlinear parabolic equations and diffusion processes
cites work: Q3992729
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cites work: Q3995203
cites work: Stochastic flows acting on Schwartz distributions
cites work: Generalized solutions of a stochastic partial differential equation
cites work: On certain systems of parabolic equations
full work available at URL: s0219493712003523
OpenAlex ID: W2078344231
title: Generalized solutions of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes
review text: The main purpose of this paper is to construct both strong and weak solutions of the Cauchy problem for a class of systems of nonlinear parabolic equations. The authors give a stochastic interpretation of such system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, the authors use some results from Kunita theory of stochastic flows applied to the case of systems of parabolic equations. Firstly the authors develop a stochastic approach to the construction of weak solution to the Cauchy problem for a system of linear parabolic equations and then extend these results to the system of nonlinear parabolic equations.
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 6042627