2021/05/07 by Kyeong-Hun Kim, Kim, Kyeong-Hun, Daehan Park +3 · 2 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #26A33 #35R60 #47G20 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2105.03013
openalex publication_date 2021/05/07 · openalex created_date 2022/11/10 · openalex updated_date 2026/07/28
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^∞ ∫0t gk(u) dwsk, tgt;0, x∈ ℝd; u(0,⋅)=u0 as well as the SPDE driven by space-time white noise ∂αtu=ϕ(Δ)u + f(u) + ∂β-1th(u) W, tgt;0,x∈ ℝd; u(0,⋅)=u0. 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 ∂tγ denotes the Caputo fractional derivative if γ>0 and the Riemann-Liouville fractional integral if γ≤0. The the spatial non-local operator ϕ(Δ) is a type of integro-differential operator whose symbol is -ϕ(|ξ|2), where ϕ is a Bernstein function satisfying \beginequation* κ0((R)/(r))^δ0 ≤ (ϕ(R))/(ϕ(r)), ∀ 00 and δ0∈ (0,1]. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.