Probabilistic Approach to Numerical Solution of the Cauchy Problem for Nonlinear Parabolic Equations

2021 ◽  
pp. 557-604
Author(s):  
Grigori N. Milstein ◽  
Michael V. Tretyakov
Keyword(s):  
Cauchy Problem ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Probabilistic Approach ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS AND DIFFUSION PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150001 ◽  
Author(s):  
YANA BELOPOLSKAYA ◽  
WOJBOR A. WOYCZYNSKI
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Diffusion Processes ◽  
Stochastic Approach ◽  
Generalized Solution ◽  
Nonlinear Parabolic Equations ◽  
Kolmogorov Equation ◽  
Generalized Solutions ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a 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, we use results from Kunita's theory of stochastic flows.

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