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Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity

Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low... J. Evol. Equ. Journal of Evolution © 2019 Springer Nature Switzerland AG Equations https://doi.org/10.1007/s00028-019-00532-6 Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity Vladimir I. Bogachev and Stanislav V. Shaposhnikov Abstract. We prove a formula representing solutions to parabolic Fokker–Planck–Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO , we show that the solution density is locally integrable to any power. 1. Introduction In the classical theory of second-order elliptic and parabolic equations, a number of integral convolution-type representations of solutions are known that involve under the integral sign suitable fundamental solutions and the represented solution itself. Such self-representations are useful in many respects, for example, in deriving various esti- mates for solution densities and proving their continuity. In case of smooth coefficients, such representations are typically obtained by multiplying the regarded equation with variable coefficients by the fundamental solution for an equation with constant coeffi- cients (that admits an explicit expression) and integrating by parts (below we consider an example of this type). However, in case of coefficients of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity

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References (25)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-019-00532-6
Publisher site
See Article on Publisher Site

Abstract

J. Evol. Equ. Journal of Evolution © 2019 Springer Nature Switzerland AG Equations https://doi.org/10.1007/s00028-019-00532-6 Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity Vladimir I. Bogachev and Stanislav V. Shaposhnikov Abstract. We prove a formula representing solutions to parabolic Fokker–Planck–Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO , we show that the solution density is locally integrable to any power. 1. Introduction In the classical theory of second-order elliptic and parabolic equations, a number of integral convolution-type representations of solutions are known that involve under the integral sign suitable fundamental solutions and the represented solution itself. Such self-representations are useful in many respects, for example, in deriving various esti- mates for solution densities and proving their continuity. In case of smooth coefficients, such representations are typically obtained by multiplying the regarded equation with variable coefficients by the fundamental solution for an equation with constant coeffi- cients (that admits an explicit expression) and integrating by parts (below we consider an example of this type). However, in case of coefficients of

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Journal of Evolution Equations – Springer Journals

Published: Aug 21, 2019

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