A sharp Lp-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

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Abstract

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) du=(aij(ω,t)uxixj+f)dt+(σik(ω,t)uxi+gk)dwtk,u(0,x)=u0, where {wtk:k=1,2,⋯} is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω,t) and can be unbounded and fully degenerate, that is, coefficients aij, σik merely satisfy [Formula presente] In this article, we prove that there exists a unique solution u to (0.1), and ‖uxxHpγ(τ,δ)≤N(d,p)(‖u0Bpγ+2(1−1/p)+‖f‖Hpγ(τ,δ1−p)+‖gxHpγ(τ,|σ|pδ1−p,l2)p+‖gxHpγ(τ,δ1−p/2,l2)), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of αij(ω,t), Hpγ(τ,δ) is a weighted stochastic Sobolev space, and Bpγ+2(1−1/p) is a stochastic Besov space.

Original languageEnglish
Pages (from-to)260-298
Number of pages39
JournalJournal of Differential Equations
Volume371
DOIs
Publication statusPublished - 2023 Oct 25

Bibliographical note

Funding Information:
I. Kim has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003959).K. Kim has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354).

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

  • Degenerate stochastic partial differential equations
  • Maximal L-regularity theory
  • Unbounded coefficients

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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