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

Kyeong Hun Kim; Daehan Park

doi:10.1007/s10959-023-01263-8

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

Kyeong Hun Kim
, Daehan Park^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

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 (Formula presented.) given with nonzero initial data. Here, ∂_t^α and ∂_t^β are the Caputo fractional derivatives, (Formula presented.) and {Z_t^k: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 (Formula presented.) 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
Pages (from-to)	671-720
Number of pages	50
Journal	Journal of Theoretical Probability
Volume	37
Issue number	1
DOIs	https://doi.org/10.1007/s10959-023-01263-8
Publication status	Published - 2024 Mar

Bibliographical note

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

Keywords

35R60
45D05
60H15
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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10.1007/s10959-023-01263-8

Cite this

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title = "A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by L{\'e}vy Processes",

abstract = "We present an Lp-theory (p≥2) for semi-linear time-fractional stochastic partial differential equations driven by L{\'e}vy processes of the type (Formula presented.) given with nonzero initial data. Here, ∂tα and ∂tβ are the Caputo fractional derivatives, (Formula presented.) and \{Ztk:k=1,2,…\} is a sequence of independent L{\'e}vy 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 (Formula presented.) where Z˙t is a multi-dimensional L{\'e}vy 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.",

keywords = "35R60, 45D05, 60H15, L{\'e}vy processes, Maximal L-regularity, Stochastic partial differential equations, Time-fractional derivatives",

author = "Kim, \{Kyeong Hun\} and Daehan Park",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.",

year = "2024",

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doi = "10.1007/s10959-023-01263-8",

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T1 - A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by Lévy Processes

AU - Kim, Kyeong Hun

AU - Park, Daehan

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.

PY - 2024/3

Y1 - 2024/3

N2 - We present an Lp-theory (p≥2) for semi-linear time-fractional stochastic partial differential equations driven by Lévy processes of the type (Formula presented.) given with nonzero initial data. Here, ∂tα and ∂tβ are the Caputo fractional derivatives, (Formula presented.) 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 (Formula presented.) 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.

AB - We present an Lp-theory (p≥2) for semi-linear time-fractional stochastic partial differential equations driven by Lévy processes of the type (Formula presented.) given with nonzero initial data. Here, ∂tα and ∂tβ are the Caputo fractional derivatives, (Formula presented.) 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 (Formula presented.) 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.

KW - 35R60

KW - 45D05

KW - 60H15

KW - Lévy processes

KW - Maximal L-regularity

KW - Stochastic partial differential equations

KW - Time-fractional derivatives

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DO - 10.1007/s10959-023-01263-8

M3 - Article

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SN - 0894-9840

VL - 37

SP - 671

EP - 720

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 1

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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