随机Coleman-Gurtin方程统计稳态的存在性与更高正则性
Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation
AI总结：
本文研究受加性高斯噪声扰动的带记忆半线性Volterra方程，证明其存在不变测度，并通过引入适应记忆核的函数空间恢复紧性，进而利用Lyapunov结构证明统计稳态具有更高正则性。
摘要：
我们研究一类具有多项式型势的半线性微分Volterra方程，该方程在受到加性高斯噪声随机扰动的同时，也纳入了记忆效应的影响。我们证明，对于一大类非线性势，该系统总是承认不变概率测度。然而，记忆效应的存在阻碍了以典型方式获得紧性。在本文中，这一障碍通过引入适应于记忆核的函数空间来克服，从而得以恢复紧性。在噪声足够光滑的假设下，进一步证明统计稳态具有由非线性结构决定的更高阶正则性。这是通过一个控制论证建立的，该论证以一种新颖的方式利用系统潜在的Lyapunov结构，将正则性渐近地转移到解上。
英文摘要：
We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.
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