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Probabilistic representation formula for the solution of fractional high-order heat-type equations

  • Stefano BonaccorsiEmail author
  • Mirko D’Ovidio
  • Sonia Mazzucchi
Article
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Abstract

We propose a probabilistic construction for the solution of a general class of fractional high-order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order \(\alpha \in (0,1)\).

Keywords

Partial differential equations Probabilistic representation of solutions of PDEs Stochastic processes 

Mathematics Subject Classification

35C15 60G50 60G20 60F05 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Stefano Bonaccorsi
    • 1
    Email author
  • Mirko D’Ovidio
    • 2
  • Sonia Mazzucchi
    • 1
  1. 1.Dipartimento di MatematicaUniversity of TrentoPovo (Trento)Italy
  2. 2.Dipartimento di Scienze di Base e Applicate per l’IngegneriaSapienza University of RomeRomeItaly

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