2022/04/30 by Kristin Kirchner, Kirchner, Kristin, Joshua S. Willems +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #35R11 (Secondary) #60G15 #60H15 (Primary) 47D06 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2205.00248
openalex publication_date 2022/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A new class of fractional-order stochastic evolution equations of the form (∂t + A)γX(t) = WQ(t), t∈[0,T], γ∈ (0,∞), is introduced, where -A generates a C0-semigroup on a separable Hilbert space H and the spatiotemporal driving noise WQ 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 := Lβ 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.