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

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

Kirchner, K. (TU Delft Analysis)
Willems, J. (TU Delft Analysis)

2023

A new class of fractional-order parabolic stochastic evolution equations of the form (∂t+A)γX(t)=W˙Q(t) , t∈ [0 , T] , γ∈ (0 , ∞) , is introduced, where - A generates a C -semigroup on a separable Hilbert space H and the spatiotemporal driving noise 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^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.

Matérn covariance
Mean-square differentiability
Mild solution
Nonlocal space–time differential operators
Spatiotemporal Gaussian processes
Strongly continuous semigroups

http://resolver.tudelft.nl/uuid:3a1578ce-3739-4dba-bc90-2bcc4105c5c6

https://doi.org/10.1007/s40072-023-00316-7

2194-0401

Stochastic Partial Differential Equations: analysis and computations

Institutional Repository

journal article

© 2023 K. Kirchner, J. Willems