Superposition principle for non-local Fokker–Planck–Kolmogorov operators

Michael Röckner; Longjie Xie; Xicheng Zhang

doi:10.1007/s00440-020-00985-8

Superposition principle for non-local Fokker–Planck–Kolmogorov operators

Michael Röckner, Longjie Xie, Xicheng Zhang^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

Abstract

We prove the superposition principle for probability measure-valued solutions to non-local Fokker–Planck–Kolmogorov equations, which in turn yields the equivalence between martingale problems for stochastic differential equations with jumps and such non-local partial differential equations with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.

Original language	English
Pages (from-to)	699-733
Number of pages	35
Journal	Probability Theory and Related Fields
Volume	178
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-020-00985-8
Publication status	Published - 1 Dec 2020
Externally published	Yes

Keywords

Fractional porous media equation
Martingale problem
Non-local Fokker–Planck–Kolmogorov equation
Superposition principle

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10.1007/s00440-020-00985-8

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abstract = "We prove the superposition principle for probability measure-valued solutions to non-local Fokker–Planck–Kolmogorov equations, which in turn yields the equivalence between martingale problems for stochastic differential equations with jumps and such non-local partial differential equations with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.",

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AU - Zhang, Xicheng

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N2 - We prove the superposition principle for probability measure-valued solutions to non-local Fokker–Planck–Kolmogorov equations, which in turn yields the equivalence between martingale problems for stochastic differential equations with jumps and such non-local partial differential equations with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.

AB - We prove the superposition principle for probability measure-valued solutions to non-local Fokker–Planck–Kolmogorov equations, which in turn yields the equivalence between martingale problems for stochastic differential equations with jumps and such non-local partial differential equations with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.

KW - Fractional porous media equation

KW - Martingale problem

KW - Non-local Fokker–Planck–Kolmogorov equation

KW - Superposition principle

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ER -

Superposition principle for non-local Fokker–Planck–Kolmogorov operators

Abstract

Keywords

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