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 $({\partial }_t+A{)}^{\gamma }X(t)={\dot{W}}^Q(t)$, $t\in [0,T]$, $\gamma \in (0,\infty )$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert space $H$ and the spatiotemporal driving noise ${\dot{W}}^Q$ 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 $Q:={\tilde{L}}^{-\alpha }$ 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-)Matérn fields to space-time.

41 pages. To appear in Stochastics and Partial Differential Equations: Analysis and Computations

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• Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping
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Cited by in corpus (3)

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