Asymptotic Analysis of Diffusion Processes with Applications to Natural Sciences

这个项目的主要目标是研究一大类问题，在这些问题中，渐近分析允许详细研究多尺度现象。所要研究的问题包括分支扩散过程的渐近行为，胞流传输的平均化和齐次化之间的转换，以及非线性抛物型偏微分方程解的渐近性。还将研究与不可压缩流动的随机扰动有关的几个问题。这些研究包括Benard对流的输运性质，随机扰动不可压缩流动的大偏差，以及布朗马达的数学分析。特别是，我们将考虑分支扩散过程，这在各种种群的进化研究中是核心的，例如细菌、癌细胞、特定基因的携带者等，其中一个种群的每个成员可能独立于其他成员死亡或产生后代。我们计划描述种群在不同空间区域的长期行为，除了分支外，种群的成员在空间中移动，分支机制取决于位置。要考虑的其他模型描述了由于宏观运动和小随机扩散的组合而导致的粒子(例如，分子)的运动。特别是，我们将研究从随机或周期速度场的波动中产生定向运动的机制，即使在场本身没有首选方向的情况下也是如此。这种机制在许多应用中非常重要，已经在数百篇物理和化学论文中进行了讨论，尽管主要是在计算和实验层面上。

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.