### 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 language | English (US) |
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Title of host publication | Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016 |

Editors | Gerald Trutnau, Andreas Eberle, Walter Hoh, Moritz Kassmann, Martin Grothaus, Wilhelm Stannat |

Publisher | Springer New York LLC |

Pages | 201-211 |

Number of pages | 11 |

ISBN (Print) | 9783319749280 |

DOIs | |

State | Published - Jan 1 2018 |

Event | International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 - Bielefeld, Germany Duration: Oct 10 2016 → Oct 14 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 229 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 |
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Country | Germany |

City | Bielefeld |

Period | 10/10/16 → 10/14/16 |

### Keywords

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

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## Cite this

*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