A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives

In this article, we present an Lp-theory (p ≥ 2) for the semi-linear stochastic partial differential equations (SPDEs) of type ∂ _t ^α u = L(ω,t,x)u +f (u)+∂ _t ^β ∞ ∑ k=1 ∫ ₀ ^t (∧^k (ω,t,x)u +g^k(u))dw _t ^k, where α ∈ (0, 2), β < α + 1/2 and ∂ _t ^α and ∂ _t ^β denote the Caputo derivatives of order α and β, respectively. The processes w _t ^k, k ∈ N = 1, 2, . . ., are independent one-dimensional Wiener processes, L is either divergence or nondivergence-type second-order operator, and ∧^k are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal L_p-regularity of the solutions. By converting SPDEs driven by d-dimensional space-time white noise into the equations of above type, we also obtain an L_p-theory for SPDEs driven by space-time white noise if the space dimensiond < 4-2(2β ^-1)α^-1. In particular, if β < 1/2 + α/4 then we can handle space-time white noise driven SPDEs with space dimension d = 1,2,3.