A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators

Kyeong Hun Kim, Daehan Park, Junhee Ryu

Research output: Contribution to journalArticlepeer-review

Abstract

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes ∂tαu=(ϕ(Δ)u+f(u))+∂tβ∑k=1∞∫0tgk(u)dwsk,t>0,x∈Rdas well as the SPDE driven by space-time white noise ∂tαu=ϕ(Δ)u+f(u)+∂tβ-1h(u)W˙,t>0,x∈Rd.Here, α∈ (0 , 1) , β< α+ 1 / 2 , {wtk:k=1,2,…} is a family of independent one-dimensional Wiener processes and W˙ is a space-time white noise defined on [0 , ∞) × Rd. The time non-local operator ∂tα denotes the Caputo fractional derivative of order α, the function ϕ is a Bernstein function, and the spatial non-local operator ϕ(Δ) is the integro-differential operator whose symbol is - ϕ(| ξ| 2). In other words, ϕ(Δ) is the infinitesimal generator of the d-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.

Original languageEnglish
Article number57
JournalJournal of Evolution Equations
Volume22
Issue number3
DOIs
Publication statusPublished - 2022 Sep

Keywords

  • Maximal L-regularity
  • Sobolev space theory
  • Space-time non-local operators
  • Space-time white noise
  • Stochastic partial differential equations

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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