Asymptotic Problems for Stochastic Processes and PDE's

考虑了几类渐近问题。确定性和随机扰动的平均原理，反应扩散方程的渐近问题，以及与随机共振有关的问题都在其中。具有守恒律的扰动系统的长期演化，即使在纯确定性扰动的情况下，通常也会导致由守恒律定义的复杂系统上的随机过程。因此，经典的平均原理(例如，当哈密顿量有许多临界点时，对于可积哈密顿系统的确定性扰动)应该在随机框架下处理。在这项研究中，研究了不可压缩二维流体中反应扩散的小扩散渐近，它与哈密顿系统的平均密切相关。另一类问题涉及大偏差理论和随机共振。考虑了自治系统中小噪声引起的大幅度振荡和镇定等新效应，渐近方法是应用数学中最有力的工具之一。特别是，当考虑组合多尺度过程的系统时，平均原理起主导作用。这样的问题出现在力学、材料科学、生物物理学和其他领域。这项研究不仅考虑了平均化原理的数学证明问题，还描述了新的应用和新的影响。近年来，最早出现在气候长时间演化理论中的随机共振型效应引起了物理学、工程学和生物学等领域专家的关注。这些效应的数学理论是

Several classes of asymptotic problems are considered. The averaging principle for deterministic and stochastic perturbations, asymptotic problems for reaction-diffusion equations, and problems related to stochastic resonance are among them. The long-time evolution of perturbed systems with conservation laws, even in the case of purely deterministic perturbations, leads, in general, to stochastic process on complexes defined by the conservation laws. So the classical averaging principle (say, for deterministic perturbations of integrable Hamiltonian systems when the Hamiltonian has many critical points) should be treated in the stochastic framework. In this research small diffusion asymptotics for reaction-diffusion in an incompressible 2D-fluid, which is closely related to the averaging for Hamiltonian systems, is studied. Another class of problems concerns the large deviation theory and stochastic resonance. A number of new effects such as large amplitude oscillations and stabilization induced by the small noise in autonomous systems are considered.The asymptotic approach is one of the most powerful tools of applied mathematics. In particular, the averaging principle plays the leading role when systems combining multi-scale processes are considered. Such problems arise in mechanics, in material sciences, in biophysics, and in other areas. This research does not just consider problems concerning the mathematical justification of the averaging principle, but also describes new applications and new effects. In recent years stochastic-resonance-type effects, which first appeared in the theory of long-time evolution of the climate, have attracted the attention of specialists in many areas of physics, engineering, and biology. The mathematical theory of these effects is