Space-time fractional stochastic partial differential equations with Lévy noise

Xiangqian Meng; Erkan Nane

doi:10.1515/fca-2020-0009

Article

Space-time fractional stochastic partial differential equations with Lévy noise

Xiangqian Meng and Erkan Nane

Published/Copyright: February 27, 2020

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From the journal Fractional Calculus and Applied Analysis Volume 23 Issue 1

Abstract

We consider non-linear time-fractional stochastic heat type equation

∂βu∂tβ+ν(−Δ)α/2u=It1−β[∫Rdσ(u(t,x),h)N~⋅(t,x,h)]

and

∂βu∂tβ+ν(−Δ)α/2u=It1−β[∫Rdσ(u(t,x),h)N⋅(t,x,h)]

in (d + 1) dimensions, where α ∈ (0, 2] and d < min{2, β⁻¹}α, ν > 0, ∂tβ is the Caputo fractional derivative, −(−Δ)^α/2 is the generator of an isotropic stable process, It1−β is the fractional integral operator, N(t, x) are Poisson random measure with Ñ(t, x) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.

MSC 2010: Primary 60H15; Secondary 35R60; 60H40

Key Words and Phrases: time-fractional stochastic partial differential equations; fractional Duhamel’s principle; Caputo derivatives; Walsh isometry

Acknowledgements

The authors thank the two anonymous referees for reading the paper carefully and for the many comments that improved the paper. The Authors also thank the editor for very useful comments that improved the paper immensely.

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Received: 2019-02-26

Revised: 2020-01-20

Published Online: 2020-02-27

Published in Print: 2020-02-25

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https://doi.org/10.1515/fca-2020-0009

Keywords for this article

time-fractional stochastic partial differential equations; fractional Duhamel’s principle; Caputo derivatives; Walsh isometry