Asymptotic Problems for Stochastic Processes & PDE's

9504177 Freidlin摘要我们研究具有守恒律的动力系统的随机扰动。第一积分的演化导致与第一积分对应的图上的扩散过程。这允许用一个小参数来解决许多偏微分方程组问题。以雷诺数较大的二维N-S方程为例，就属于这类问题。我们还研究了扩散-变形过程的大偏差。这些结果有助于描述反应扩散方程中波阵面和其他模式的渐近行为。实际过程的数学模型有时被证明过于复杂，难以分析。但与其他参数相比，模型中包含的一些参数往往较小(或较大)。这使得我们可以简化模型，并经常以便于应用的形式获得解。我们研究了动力系统的小随机扰动、反应扩散模型的大尺度逼近以及与随机因素和确定性因素相互作用有关的其他问题。

9504177 Freidlin Abstract We study random perturbations of dynamical systems with conservation laws. The evolution of the first integral leads to a diffusion process on a graph corresponding to the first integral. This allows progress in a number of PDE problems with a small parameter. The two-dimensional Navier-Stocks equation with a large Reynolds number, for exampke, belongs to this class of problems. We study also large deviations for the diffusion-transmutation processes. These results help to describe the asymptotic behavior of wave fronts and other patterns in reaction-diffusion equations. Mathematical models of real processes sometimes,turn out to be too complicated for analysis. But very often some of the parameters included in the model are small (or large) in comparison with the others. This allows us to simplify the model and often to obtain the solution in a form convenient for applications. We study small random perturbations of dynamical systems, large scale approximation in reaction- diffusion models and other problems related to an interplay between random and deterministic factors.