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

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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Acknowledgements

The authors are very grateful to the referees for their quite useful suggestions.

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Correspondence to Xicheng Zhang.

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This work is supported by NNSF of China (Nos. 11731009, 11931004), NSF of Jiangsu (BK20170226) and the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”.

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Röckner, M., Xie, L. & Zhang, X. Superposition principle for non-local Fokker–Planck–Kolmogorov operators. Probab. Theory Relat. Fields 178, 699–733 (2020). https://doi.org/10.1007/s00440-020-00985-8

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  • DOI: https://doi.org/10.1007/s00440-020-00985-8

Keywords

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

Mathematics Subject Classification

  • 60H10
  • 60J75
  • 40K05