arXiv subjects

Joshua Willems

Publications and source records attributed to Joshua Willems.

4 recordsLinked to original sources

arXiv2025

Discrete-to-continuum limits of semilinear stochastic evolution equations in Banach spaces

We study the convergence of semilinear parabolic stochastic evolution equations, posed on a sequence of Banach spaces approximating a limiting space and driven by additive white noise projected onto the former spaces. Under appropriate uniformity and convergence conditions on the linear operators, nonlinear drifts and initial data, we establish convergence of the associated mild solution processes when lifted to a common state space. Our framework is applied to the case where the limiting problem is a stochastic partial differential equation whose linear part is a generalized Whittle-Mat\'ern operator on a manifold $\mathcal{M}$, discretized by a sequence of graphs constructed from a (random) point cloud. In this setting we obtain various discrete-to-continuum convergence results for solutions lifted to $L^q(\mathcal{M})$ for $q \in [2,\infty]$, one of which recovers the $L^\infty$-convergence of a finite-difference discretization of certain (fractional) stochastic Allen-Cahn equations.

math.PR↗

arXiv2024

Dirichlet problems associated to abstract nonlocal space-time differential operators

Let the abstract fractional space-time operator $(\partial_t + A)^s$ be given, where $s \in (0,\infty)$ and $-A \colon \mathsf{D}(A) \subseteq X \to X$ is a linear operator generating a uniformly bounded strongly measurable semigroup $(S(t))_{t\ge0}$ on a complex Banach space $X$. We consider the corresponding Dirichlet problem of finding a function $u \colon \mathbb{R} \to X$ such that $(\partial_t + A)^s u(t) = 0$ on $(t_0, \infty)$ and $u(t) = g(t)$ on $(-\infty, t_0]$, for given $t_0 \in \mathbb{R}$ and $g \colon (-\infty,t_0] \to X$. We define the concept of $L^p$-solutions, to which we associate a mild solution formula which expresses $u$ in terms of $g$ and $(S(t))_{t\ge0}$ and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem $u' + Au = 0$ on $(t_0, \infty)$ with $u(t_0) = x \in \overline{\mathsf{D}(A)}$. Moreover, we include a comparison to analogous solution concepts arising from Riemann-Liouville and Caputo type initial value problems.

math.AP↗

arXiv2023

Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations

We define various higher-order Markov properties for stochastic processes $(X(t))_{t\in \mathbb{T}}$, indexed by an interval $\mathbb{T} \subseteq \mathbb{R}$ and taking values in a real and separable Hilbert space $U$. We furthermore investigate the relations between them. In particular, for solutions to the stochastic evolution equation $\mathcal{L} X = \dot W^Q\!$, where $\mathcal{L}$ is a linear operator acting on functions mapping from $\mathbb{T}$ to $U$ and $(\dot W^Q(t))_{t\in\mathbb{T}}$ is the formal derivative of a $U$-valued (cylindrical) $Q$-Wiener process, we prove necessary and sufficient conditions for the weakest Markov property via locality of the precision operator $\mathcal{L}^*\! \mathcal{L}$. As an application, we consider the space-time fractional parabolic operator $\mathcal{L} = (\partial_t + A)^\gamma$ of order $\gamma \in (1/2,\infty)$, where $-A$ is a linear operator generating a $C_0$-semigroup on $U$. We prove that the resulting solution process satisfies an $N$th order Markov property if $\gamma = N \in \mathbb{N}$ and show that a necessary condition for the weakest Markov property is generally not satisfied if $\gamma \notin \mathbb{N}$. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle-Mat\'ern Gaussian random fields if $U = L^2(\mathcal{D})$ for a spatial domain $\mathcal{D}\subseteq\mathbb{R}^d\!$. Secondly, we show that a $U$-valued analog to the fractional Brownian motion with Hurst parameter $H \in (0,1)$ can be obtained as the limiting case of $\mathcal{L} = (\partial_t + \varepsilon \, \mathrm{Id}_U)^{H + \frac{1}{2}}$ for $\varepsilon \downarrow 0$.

math.PR↗

arXiv2022

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

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-)Mat\'ern fields to space-time.

math.PR↗