AI 中文总结
本文针对具有Hölder扩散系数的随机时间非局部电报方程，通过遗传相空间提升框架与新型广义耦合方法，首次证明其弱解的存在性与律唯一性，还将该框架拓展至低正则性非线性项的处理。
AI 中文摘要
我们考虑具有$(\boldsymbol{\text{PC}}_\boldsymbol{\boldsymbol{\text{ε}}})$型核函数$a$的随机时间非局部电报方程的初边值问题：$\boldsymbol{\text{γ} \text{∂}_t (a \boldsymbol{*} \text{∂}_t (a \boldsymbol{*} v)) = Δ v - \text{∂}_t (a \boldsymbol{*} v) + \boldsymbol{\text{Ψ}}(v) + \boldsymbol{\text{Φ}}(v) \frac{\text{d}W(t)}{\text{d}t}}$，其中$W$是时空高斯白噪声，$\boldsymbol{\text{Ψ}}$满足线性增长条件，$\boldsymbol{\text{Φ}}$是Hölder连续且一致非退化的。该模型刻画了小尺度系统在随机波动下的高频信号传播。我们为时间非局部电报方程开发了新的遗传相空间提升框架，还提出了一种新型广义耦合框架，其特点是对速度项构造了新的阻尼控制项。基于这些分析工具，我们首次证明了取值于$\boldsymbol{L}_{\text{loc}}^2(\boldsymbol{\text{R}}_+; \boldsymbol{H}^{\boldsymbol{\text{δ}}})$的随机非局部电报方程的弱存在性与律唯一性，其中正则性指数$\boldsymbol{\text{δ}}$可从下方任意接近$\boldsymbol{\text{min}}\boldsymbol{\bigg\rbrace\frac{1}{2},\frac{\boldsymbol{\text{ε}}}{2+\boldsymbol{\text{ε}}}\bigg\rbrace$，且Hölder指数$\boldsymbol{\text{κ}}$的容许下界由可积性指数$\boldsymbol{\text{ε}}$定量确定。对于将$a$替换为狄拉克测度$\boldsymbol{\text{δ}}_0$得到的随机阻尼波动方程对应的初边值问题，$\boldsymbol{\text{κ}}$可提升至$\boldsymbol{(\frac{3}{5}, 1]}$内的任意值。更重要的是，该广义耦合框架还能处理同时依赖位移和速度的低正则性非线性项。
英文摘要
We consider the initial-boundary value problem for the stochastic time-nonlocal telegraph equation with $(\mathcal{PC}_\varepsilon)$-type kernel $a$: \begin{align*} γ\partial_t \left( a \ast \partial_t (a \ast v)\right) =Δv-\partial_t (a \ast v)+ Ψ(v)+ Φ(v) \frac{\mathrm{d}W(t)}{\mathrm{d}t}, \end{align*} where $W$ is a space-time Gaussian white noise, $Ψ$ satisfies a linear growth condition, and $Φ$ is Hölder continuous and uniformly nondegenerate. This model characterizes high-frequency signal propagation in small-scale systems under stochastic fluctuations. We develop a new hereditary phase-space lifting framework for time-nonlocal telegraph equations. In addition, we propose a novel generalized coupling framework, which features a new construction of the damping control term for the velocity. Based on these analytic tools, we prove the first results on weak existence and uniqueness in law for mild solutions, valued in $L_{loc}^2(\mathbb R_+; H^δ)$, to the stochastic nonlocal telegraph equation. The regularity index $δ$ can be arbitrarily close to $\min\{\frac{1}{2},\frac{\varepsilon}{2+\varepsilon} \}$ from below, and the admissible lower bound of the Hölder exponent $κ$ is quantitatively determined by the integrability exponent $\varepsilon$. For the corresponding IBVP of the stochastic damped wave equation, obtained by replacing $a$ with the Dirac measure $δ_0$, $κ$ can be improved to any value in $(\frac{3}{5}, 1]$. More significantly, the generalized coupling framework also handles low-regularity nonlinearities depending on both displacement and velocity.
