A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by Lévy Processes

Kyeong Hun Kim, Daehan Park

Research output: Contribution to journalArticlepeer-review

Abstract

We present an Lp -theory (p≥ 2) for semi-linear time-fractional stochastic partial differential equations driven by Lévy processes of the type ∂tαu=∑i,j=1daijuxixj+f(u)+∑k=1∞∂tβ∫0t(∑i=1dμikuxi+gk(u))dZsk given with nonzero initial data. Here, ∂tα and ∂tβ are the Caputo fractional derivatives, 0<α<2,β<α+1/p, and {Ztk:k=1,2,…} is a sequence of independent Lévy processes. The coefficients are random functions depending on (t, x). We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity of the solution. As an application, we also obtain an Lp -regularity theory of the equation ∂tαu=∑i,j=1daijuxixj+f(u)+∂tβ∫0th(u)dZs, where Z˙ t is a multi-dimensional Lévy space-time white noise with space dimension d<4-2(2β-2/p)+α . In particular, if β< α/ 4 + 1 / p , then one can take d= 1 , 2 , 3 .

Original languageEnglish
JournalJournal of Theoretical Probability
DOIs
Publication statusAccepted/In press - 2023

Bibliographical note

Funding Information:
The authors were supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Lévy processes
  • Maximal L-regularity
  • Stochastic partial differential equations
  • Time-fractional derivatives

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty

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