## Abstract

We present an L_{p} -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 L_{p} -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 language | English |
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Journal | Journal of Theoretical Probability |

DOIs | |

Publication status | Accepted/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