Poisson stochastic process and basic schauder and Sobolev estimates in the theory of parabolic equations (short version)

N. V. Krylov, E. Priola

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. We only give examples of applications of our results. Their proofs will appear elsewhere.

Original languageEnglish (US)
Title of host publicationStochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016
EditorsGerald Trutnau, Andreas Eberle, Walter Hoh, Moritz Kassmann, Martin Grothaus, Wilhelm Stannat
PublisherSpringer New York LLC
Pages201-211
Number of pages11
ISBN (Print)9783319749280
DOIs
StatePublished - Jan 1 2018
EventInternational conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 - Bielefeld, Germany
Duration: Oct 10 2016Oct 14 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume229
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherInternational conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016
CountryGermany
CityBielefeld
Period10/10/1610/14/16

Keywords

  • Multidimensional parabolic equations
  • Poisson process
  • Schauder estimates
  • Sobolev-space estimates

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  • Cite this

    Krylov, N. V., & Priola, E. (2018). Poisson stochastic process and basic schauder and Sobolev estimates in the theory of parabolic equations (short version). In G. Trutnau, A. Eberle, W. Hoh, M. Kassmann, M. Grothaus, & W. Stannat (Eds.), Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016 (pp. 201-211). (Springer Proceedings in Mathematics and Statistics; Vol. 229). Springer New York LLC. https://doi.org/10.1007/978-3-319-74929-7_10