Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation
Abstract
We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain Ocal⊂Rd driven by additive trace-class noise, where the elastic response is governed by a \emph{mixed local--nonlocal} operator Acal=−θΔ+β(−Δ)s with s∈(0,1). A fundamental challenge in this setting is that the local and nonlocal operators do not commute on bounded domains: the natural Dirichlet basis fails to diagonalize the restricted fractional Laplacian. Consequently, we first establish the \emph{strong} convergence of the resulting non-diagonal, dense Galerkin approximation scheme. Leveraging these uniform energy bounds, we rigorously derive the associated Itô energy identity. In the defocusing regime (ε = +1), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space V × H. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing (u(t), v(t)) ∈ 𝔻1,2(V) × 𝔻1,2(H), and leverage fractional Sobolev embeddings to prove that the one-dimensional probability law of u(t, x0) is absolutely continuous via the Bouleau--Hirsch criterion. In stark contrast, for the focusing regime (ε = −1), we establish local well-posedness and prove a rigorous dichotomy: under a negativity condition on the initial energy, either pathwise explosion occurs with positive probability in finite time, or the energy norm possesses an infinite second moment before an explicit critical time T*. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.
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BibTeX
@misc{delgadovences2026strong,
title = {{Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation}},
author = {Francisco Delgado-Vences and Jose Julian Pavon-Español},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06883},
eprint = {2609.06883},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06883},
}
RIS
TY - UNPB
TI - Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation
AU - Francisco Delgado-Vences
AU - Jose Julian Pavon-Español
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06883
UR - https://arxiv.org/abs/2609.06883
AB - We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain Ocal⊂R^d driven by additive trace-class noise, where the elastic response is governed by a \emph{mixed local--nonlocal} operator Acal=−θΔ+β(−Δ)^s with s∈(0,1). A fundamental challenge in this setting is that the local and nonlocal operators do not commute on bounded domains: the natural Dirichlet basis fails to diagonalize the restricted fractional Laplacian. Consequently, we first establish the \emph{strong} convergence of the resulting non-diagonal, dense Galerkin approximation scheme. Leveraging these uniform energy bounds, we rigorously derive the associated Itô energy identity. In the defocusing regime (ε = +1), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space V × H. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing (u(t), v(t)) ∈ 𝔻^1,2(V) × 𝔻^1,2(H), and leverage fractional Sobolev embeddings to prove that the one-dimensional probability law of u(t, x_0) is absolutely continuous via the Bouleau--Hirsch criterion. In stark contrast, for the focusing regime (ε = −1), we establish local well-posedness and prove a rigorous dichotomy: under a negativity condition on the initial energy, either pathwise explosion occurs with positive probability in finite time, or the energy norm possesses an infinite second moment before an explicit critical time T^*. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.
ER -