Foundations and Applications of Stochastic Analysis

PI将研究在应用科学和数学研究启发的数学模型研究中出现的基础问题。其中一个问题，即所谓的第三边界问题，是具有半透膜的真实的系统的数学模型。PI的将发展这种系统的分形结构的数学理论。将研究离散模型对连续过程的近似质量。简单的随机过程，如布朗运动，可以以各种方式扰动，例如，通过惰性漂移或斜反射。这些更复杂的模型需要新的数学工具，这将被开发。PI的将研究一些多粒子系统局限于有界域。一些数学工具将得到改进，包括边界Harnack原理和热核估计的新版本。该项目将开发数学科学应用所需的理论数学工具，并受到数学科学应用的启发。引起正在考虑的想法的应用现象包括天然半透膜，如人类肺部和人造物体，如催化转化器和汽车电池。其中一些应用与计算机科学有关，特别是用适合于基于计算机的分析的数字模型近似真实的连续现象。PI还将研究根据各种方案描述种群进化的数学模型。

The PI's will study foundational questions that arise in the study of mathematical models inspired by applied science and mathematical research. One of the problems, the so-called third boundary problem, is a mathematical model for real systems with semipermeable membranes. The PI's will develop the mathematical theory of such systems with fractal structure. The quality of approximation of continuous processes by discrete models will be investigated. Simple stochastic processes, such as Brownian motion, can be perturbed in various ways, for example, by inert drift or oblique reflection. These more complicated models require new mathematical tools which will be developed. The PI's will study some multiparticle systems confined to bounded domains. Several mathematical tools will be sharpened, including new versions of the boundary Harnack principle and heat kernel estimates.The project will develop theoretical mathematical tools needed in and inspired by scientific application of mathematics.The applied phenomena that gave rise to the ideas under consideration include natural semipermeable membranes, such as human lungs, and man-made objects, such as catalytic converters and car batteries.Some of the applications are related to computer science, especially approximation of real continuous phenomena with digital models suitable for computer based analysis. The PI's will also study mathematical models describing populations undergoing evolution according to various schemes.