On Nonuniqueness of Probability Solutions to the Cauchy Problem for the Fokker–Planck–Kolmogorov Equation

2021 ◽  
Vol 103 (3) ◽  
pp. 108-112
Author(s):  
V. I. Bogachev ◽  
T. I. Krasovitskii ◽  
S. V. Shaposhnikov
Keyword(s):  
Cauchy Problem ◽  
Kolmogorov Equation ◽  
The Cauchy Problem ◽  
Fokker Planck

scholarly journals On the superposition principle for Fokker-Planck-Kolmogorov equations

2019 ◽  
Vol 487 (5) ◽  
pp. 483-486
Author(s):  
V. I. Bogachev ◽  
M. Röckner ◽  
S. V. Shaposhnikov
Keyword(s):  
Cauchy Problem ◽  
Superposition Principle ◽  
Martingale Problem ◽  
Kolmogorov Equation ◽  
Kolmogorov Equations ◽  
The Cauchy Problem ◽  
Fokker Planck

We give a generalization of the so-called superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem.

THE GLOBAL CLASSICAL SOLUTION OF THE VLASOV–MAXWELL–FOKKER–PLANCK SYSTEM NEAR MAXWELLIAN

2011 ◽  
Vol 21 (05) ◽  
pp. 1007-1025 ◽  
Author(s):  
MYEONGJU CHAE
Keyword(s):  
Cauchy Problem ◽  
Electromagnetic Field ◽  
Classical Solution ◽  
Charged Particles ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Self Consistent ◽  
The Cauchy Problem ◽  
Fokker Planck

The Vlasov–Maxwell–Fokker–Planck system is used in modeling distribution of charged particles in plasma, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.

The Exact Solution of the Cauchy Problem for Two Generalized Fokker-Planck Equations — Algebraic Approach

1997 ◽  
Vol 12 (01) ◽  
pp. 165-170 ◽  
Author(s):  
A. A. Donkov ◽  
A. D. Donkov ◽  
E. I. Grancharova
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Algebraic Approach ◽  
Planck Equation ◽  
Operational Method ◽  
Suzuki Method ◽  
Fokker Planck Equation ◽  
The Cauchy Problem ◽  
Fokker Planck Equations ◽  
Fokker Planck

By employing algebraic techniques we find the exact solutions of the Cauchy problem for two equations, which may be considered as n-dimensional generalization of the famous Fokker–Planck equation. Our approach is a combination of the disentangling techniques of R. Feynman with operational method developed in modern functional analysis in particular in the theory of partial differential equations. Our method may be considered as a generalization of the M. Suzuki method of solving the Fokker–Planck equation.

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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