Regularity theory for a new class of fractional parabolic stochastic evolution equations

Abstract

A new class of fractional-order stochastic evolution equations of the form (∂t + A) γ X(t) = Ẇ Q (t), t ∈ [0, T ], γ ∈ (0, ∞), is introduced, where −A generates a C 0 -semigroup on a separable Hilbert space H and the spatiotemporal driving noise Ẇ Q is an H-valued cylindrical Q-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by meansquare or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when A := L β and Q := L −α are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time.