Mathematical Sciences: Asymptotic Problems for Nonlinear PDE's and Limit Theorems for Random Procesess and Fields

主要研究人员将利用它们与马尔可夫过程的联系来研究线性和非线性偏微分方程组(PDE‘s)和偏微分方程组的一些渐近问题。将考虑半线性抛物型系统中波前、脉冲和其他结构的运动，以及具有小扩散系数的椭圆型和抛物型耦合方程。从概率的角度来看，对于相应的随机过程，其结果将是平均原则型或中心极限定理和大偏差型。研究人员还将研究演化偏微分方程组的随机扰动。研究人员计划发展一种类似于有限维系统随机扰动理论的理论。文中将考虑由扰动偏微分方程组定义的随机场的各种极限定理。主要研究人员将利用它们与马尔可夫过程的联系来研究线性和非线性偏微分方程组(PDE‘s)和偏微分方程组的一些渐近问题。这项工作结合了两个数学研究领域，偏微分方程组和概率论。

The principal investigators will study a number of asymptotic problems for linear and nonlinear partial differential equations (PDE's) and systems of PDE's using their connections with Markov processes. Such topics as motion of wave fronts, impulses and other structures in semilinear parabolic systems, coupled elliptic and parabolic equations with small diffusivity will be considered. From a probabilistic point of view the results will be of the averaging principle type or of central limit theorem and large deviation type for the corresponding random processes. The investigators also will study random perturbations of evolutionary PDE's. The investigators plan to develop a theory similar to the theory of random perturbations of finite dimensional systems. Many kinds of limit theorems for the random fields defined by the perturbed PDE's will be considered. The principal investigators will study a number of asymptotic problems for linear and nonlinear partial differential equations (PDE's) and systems of PDE's using their connections with Markov processes. This work brings together two fields of mathematical research, partial differential equations and probability theory.