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

A new class of fractional-order parabolic stochastic evolution equations of the form (partial derivative(t) +A)(gamma) X (t) = W-center dot( Q)(t), t is an element of [0, T], gamma is an element of (0, infinity), is introduced, where -A generates a C-0-semigroup on a separable Hilbert space H and the spatiotemporal driving noise W-center dot (Q) is the formal time derivative of 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 mean-square 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-beta and Q := L-alpha 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-)Mat & eacute;rn fields to space-time.