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
Repository Citation
Chen, L.,
Hu, Y.,
Nualart, D.
(2019).
Nonlinear Stochastic Time-Fractional Slow and Fast Diffusion Equations on Rd.
Stochastic Processes and their Applications
1-40.
http://dx.doi.org/10.1016/j.spa.2019.01.003
