Asymptotic Analysis of Diffusion Processes with Applications to Natural Sciences

扩散过程的渐近分析及其在自然科学中的应用

• 批准号: 1309084
• 负责人: Leonid Koralov
• 金额: $ 15.07万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309084&HistoricalAwards=false
• 关键词: Asymptotic; Analysis; Diffusion; Processes; Applications

The main goal of this project is to investigate a large class of problems in which asymptotic analysis allows for a detailed study of multi-scale phenomena. Among the problems to be studied are the asymptotic behavior of branching diffusion processes, transition between the averaging and homogenization regimes for transport by cellular flows, and asymptotics of solutions to nonlinear parabolic PDEs. Several problems concerning random perturbations of incompressible flows will also be studied. These include the transport properties of the Benard convection, large deviations for randomly perturbed incompressible flows, and mathematical analysis of Brownian motors.The proposed research will provide a rigorous mathematical foundation for several phenomena that have been actively discussed in natural sciences. In particular, we will consider branching diffusion processes, which are central in the study of evolution of various populations such as bacteria, cancer cells, carriers of a particular gene, etc., where each member of a population may die or produce offspring independently of the rest. We plan to describe the long-time behavior of the population in different regions of space when, in addition to branching, the members of the population move in space and the branching mechanism depends on the location. Other models to be considered describe the movement of particles (e.g., molecules) due to a combination of a macroscopic motion and a small random diffusion. In particular, we'll examine mechanisms that create directed motion out of fluctuations of a random or periodic velocity field even in situations when the field itself has no preferred direction. Such mechanisms are very important in many applications and have been discussed in hundreds of physics and chemistry papers, although mostly at computational and experimental levels.

该项目的主要目的是研究大量的问题，在这种问题中，渐近分析允许对多尺度现象进行详细研究。 在要研究的问题中，有分支扩散过程的渐近行为，平均和均质化方案之间通过细胞流进行运输的过渡以及对非线性抛物线PDE的溶液的渐近性。还将研究有关不可压缩流的随机扰动的几个问题。 其中包括贝纳德对流的运输特性，用于随机扰动不可压缩流的大偏差以及对布朗电动机的数学分析。拟议的研究将为在自然科学中积极讨论的几种现象提供严格的数学基础。特别是，我们将考虑分支扩散过程，这些过程在研究各种种群的进化中至关重要，例如细菌，癌细胞，特定基因的载体等，在该研究中，每个人口的每个成员都可能死亡或与其他人独立生产后代。我们计划描述在空间不同区域中人口的长期行为，除了分支外，人口成员在太空中移动，分支机理取决于位置。其他要考虑的模型描述了由于宏观运动和较小的随机扩散的组合而导致的颗粒运动（例如分子）的运动。特别是，即使在该场本身没有首选方向的情况下，即使在情况下，也会检查出从随机或周期性速度场的波动中产生定向运动的机制。这种机制在许多应用中非常重要，并且已经在数百种物理和化学论文中进行了讨论，尽管大多在计算和实验水平上。