混合局部-非局部随机波动方程的强Galerkin逼近、Malliavin正则性与爆破
Strong Galerkin Approximation, Malliavin Regularity, and Blow-Up for a Mixed Local--Nonlocal Stochastic Wave Equation
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中文总结 AI 辅助
研究混合局部-非局部随机波动方程，建立强Galerkin逼近收敛性，证明散焦情形全局适定性与Malliavin正则性，聚焦情形下给出爆炸与矩爆炸的二分法。
中文摘要 AI 辅助
我们研究了一类由加性迹类噪声驱动的、定义在有界光滑域 $\Ocal\subset\R^d$ 上的半线性随机波动方程的动力学行为，其中弹性响应由混合局部-非局部算子 $\Acal=-\theta\Delta+\beta(-\Delta)^s$（$s\in(0,1)$）控制。该设定中的一个基本挑战是：在有界域上，局部算子与非局部算子不可交换；自然的Dirichlet基无法对角化受限分数阶Laplacian。因此，我们首先建立了由此产生的非对角、稠密Galerkin逼近格式的强收敛性。利用这些一致能量界，我们严格推导了相应的Itô能量恒等式。在散焦情形（$\varepsilon = +1$）下，该强逼近在能量次临界范围内给出全局适定性，在能量空间 $V \times H$ 中提供唯一的概率强解。在此变分框架内，我们进行了Malliavin正则性分析，证明 $(u(t), v(t)) \in \mathbb{D}^{1,2}(V) \times \mathbb{D}^{1,2}(H)$，并利用分数阶Sobolev嵌入，通过Bouleau--Hirsch准则证明了 $u(t, x_0)$ 的一维概率律是绝对连续的。与此形成鲜明对比的是，在聚焦情形（$\varepsilon = -1$）下，我们建立了局部适定性并证明了一个严格二分法：在初始能量为负的条件下，要么路径wise爆炸以正概率在有限时间内发生，要么在显式临界时间 $T^*$ 之前能量范数具有无穷二阶矩。最后，我们观察到稠密Galerkin相互作用矩阵对空间统计推断构成了独特的结构性挑战。
英文摘要
We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\Ocal\subset\R^d$ 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 $s\in(0,1)$. 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 ($\varepsilon = +1$), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space $V \times H$. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing $(u(t), v(t)) \in \mathbb{D}^{1,2}(V) \times \mathbb{D}^{1,2}(H)$, 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 ($\varepsilon = -1$), 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.
发表机构
- Universitat Autònoma de Barcelona（巴塞罗那自治大学）
- Universidad Nacional Autónoma de México（墨西哥国立自治大学）
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