Asymptotic Problems for Stochastic Processes and Differential Equations

几类渐近问题将被考虑。第一类是关于多尺度系统的亚稳态和随机共振。在大偏差框架下研究亚稳性和随机共振的一般方法在我们以前的论文中得到了发展。在多尺度系统的情况下，这些概念应该被修改。特别是，新的效果出现在系统中，接近哈密顿量和系统中的一个小的延迟。另一类由偏微分方程的小参数和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，应该考虑与原始问题相关的图或开书空间上的方程。我们将考虑不可压缩流体、强各向异性介质和细管中的反应扩散对流方程。最后一类问题涉及多自由度和多阱哈密顿系统的平均原理。事实证明，即使在纯决定论系统中，慢运动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，通常可以忽略不计的系统的小扰动可以在更大的时间尺度上变得至关重要。研究这种影响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期影响，并在一定条件下，描述了系统的长期演化，这基本上取决于扰动。从物理学、生物学到经济学和社会学，这类问题在许多应用中出现。我们正在研究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.