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Poisson Stochastic Process and Basic Schauder and Sobolev Estimates in the Theory of Parabolic Equations (Short Version)

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Stochastic Partial Differential Equations and Related Fields (SPDERF 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 229))

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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.

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References

  1. Krylov, N.V.: A parabolic Littlewood–Paley inequality with applications to parabolic equations. Topol. Methods Nonlinear Anal.; J. Juliusz Schauder Cent. 4(2), 355–364 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Krylov, N.V.: On \(L_{p}\)-theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27(2), 313–340 (1996)

    Article  MathSciNet  Google Scholar 

  3. Krylov, N.V.: Introduction to the Theory of Random Processes. American Mathematical Society, Providence (2002)

    Book  Google Scholar 

  4. Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  5. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear parabolic equations, Nauka, Moscow (1967), in Russian; English translation: American Mathematical Society, Providence, RI (1968)

    Google Scholar 

  6. Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge (1996)

    Book  Google Scholar 

  7. Priola, E.: \(L^p\)-parabolic regularity and non-degenerate Ornstein-Uhlenbeck type operators. In: Citti, G., et al. (eds.) Geometric Methods in PDEs. Springer INdAM Series, vol. 13, pp. 121–139. Springer, Berlin (2015)

    Chapter  Google Scholar 

  8. Sato, K.I.: Lévy Processes and Infinite Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

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Acknowledgements

The article is based on the talk given by the first author at the international conference “Stochastic Partial Differential Equations and Related Topics” October 10–14, 2016, Bielefeld University. The opportunity to give this talk is greatly appreciated.

Author information

Authors and Affiliations

  1. 127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455, USA

    N. V. Krylov

  2. Dipartimento di Matematica, Universita di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    E. Priola

Authors
  1. N. V. Krylov
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  2. E. Priola
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Corresponding author

Correspondence to N. V. Krylov .

Editor information

Editors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Bonn, Germany

    Andreas Eberle

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Martin Grothaus

  3. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Walter Hoh

  4. Faculty of Mathematics, Bielefeld University, Bielefeld, Germany

    Moritz Kassmann

  5. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Wilhelm Stannat

  6. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea (Republic of)

    Gerald Trutnau

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Krylov, N.V., Priola, E. (2018). Poisson Stochastic Process and Basic Schauder and Sobolev Estimates in the Theory of Parabolic Equations (Short Version). In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_10

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