论文标题
随机Coleman-gurtin热传导中长期统计的短期记忆限制
The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction
论文作者
论文摘要
我们考虑具有多项式型电势的一类半线性差异方程，该方程融合了记忆的效果，同时通过加性高斯噪声受到随机扰动的影响。我们的主要研究是随着记忆内核倒入狄拉克功能的奇异状态中系统的长期统计数据。具体而言，我们表明，规定空间中足够多的方向是随机强迫的，就有一个独特的不变概率措施，对于合适的Wasserstein-type拓扑，系统收敛于此，并且以指数级的速率与内存核的衰减率无关。然后，我们证明了这种独特的统计稳态与在零内存限制中经典随机反应扩散方程的唯一不变概率度量的收敛性。因此，我们建立了短记忆近似的全局时间有效性。
We consider 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. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.
