Title

Nonlinear Stochastic Time-Fractional Slow and Fast Diffusion Equations on Rd

Document Type

Article

Publication Date

1-30-2019

Publication Title

Stochastic Processes and their Applications

First page number:

1

Last page number:

40

Abstract

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: ((∂^β)+(ν/2)(−Δ)^α∕2)(u(t,x))=Itγ[ρ(u(t,x))Ẇ(t,x)], t>0, x∈Rd, where Ẇ is the space–time white noise, α∈(0,2], β∈(0,2), γ≥0 and ν>0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: d<2α+(α/β)min(2γ−1,0). In some cases, the initial data can be measures. When β∈(0,1], we prove the sample path regularity of the solution.

Keywords

Nonlinear stochastic fractional diffusion equations; Measure-valued initial data; Hölder continuity; Intermittency; The Fox H-function

Disciplines

Mathematics

Language

English

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