Well-posedness of stochastic time-nonlocal telegraph equations with H\"{o}lder diffusion coefficient: hereditary phase-space lifting and novel generalized coupling method

Abstract

We consider the initial-boundary value problem for the stochastic time-nonlocal telegraph equation with $(\mathcal{PC}_\varepsilon)$-type kernel $a$: \begin{align*} \gamma \partial_t \left( a \ast \partial_t (a \ast v)\right) =\Delta v-\partial_t (a \ast v)+ \Psi(v)+ \Phi(v) \frac{\mathrm{d}W(t)}{\mathrm{d}t}, \end{align*} where $W$ is a space-time Gaussian white noise, $\Psi$ satisfies a linear growth condition, and $\Phi$ is H\"{o}lder 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 $L_{loc}^2(\mathbb R_+; H^{\delta})$, to the stochastic nonlocal telegraph equation. The regularity index $\delta$ can be arbitrarily close to $\min\{\frac{1}{2},\frac{\varepsilon}{2+\varepsilon} \}$ from below, and the admissible lower bound of the H\"{o}lder exponent $\kappa$ 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 $\delta_0$, $\kappa$ 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.