Asymptotic Problems for Stochastic Processes and Differential Equations

将考虑几类渐近问题。第一类涉及多尺度系统的亚稳性和随机共振。我们在以前的文章中发展了大偏差框架下的亚稳性和随机共振的一般方法。在多尺度系统的情况下，这些概念应该被修改。特别是，在接近哈密顿量的系统和具有小延迟的系统中出现了新的效应。另一类是具有小参数的偏微分方程解及其相关的随机过程的新问题。为了描述这些经典问题中的极限行为，我们应该考虑与原始问题有关的图上的方程或开卷空间上的方程。将考虑不可压缩流体、强各向同性介质和窄管中的反应-扩散-平流方程。最后一类问题涉及具有多个自由度和多井哈密顿量的系统的平均原理。结果表明，即使在纯决定论系统中，慢动作在某种意义上也可以是随机的。当系统在短时间间隔上考虑时，系统的小扰动通常具有可以忽略的影响，但在较大的时间尺度上可能变得至关重要。对这种影响的研究是我们的主要目标之一。我们考虑了小的随机和确定性扰动的长时间影响，并在一定条件下描述了系统的长时间演化，本质上依赖于这些扰动。这类问题出现在从物理学和生物学到经济学和社会学的许多应用中。我们正在开发解决这些问题的新的数学方法。

Several classes of asymptotic problems will be considered. The first class concerns metastability and stochastic resonance for multiscale systems. The general approach to metastability and stochastic resonance in the framework of large deviations was developed in our previous papers. In the case of multiscale systems these notions should be modified. In particular, new effects appear in systems which are close to Hamiltonian and in systems with a small delay. Another class consists of new problems for PDE's with a small parameter and related stochastic processes. To describe the limiting behavior in these classical problems one should consider equations on graphs or open book spaces related to the original problam. Reaction-Diffusion-Advection equations in incompressible fluids, in strongly unisotropic media, and in narrow tubes will be considered. The last class of problems concerns the averaging principle for systems with many degrees of freedom and multiwell Hamiltonians. It turns out that even in pure deteministic systems the slow motion can be, in a sense, stochastic.Small perturbations of a system which often have negligible effects, when the system considered on a short time interval, can become crucial on larger timescale. The study of such effects is one of our main goals. We consider long time influence of small stochastic and deterministic perturbations and, under certain conditions, describe the long-time evolution of the system which essentially depends on the perturbations. Such kind of problems arise in numerous applications from physics and biology to economics and sociology. We are developing new mathematical approches to these problems.