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，其中渐近分析允许对多尺度现象进行详细的研究。研究的问题包括分支扩散过程的渐近行为，细胞流输运的平均和均匀化区域之间的过渡，以及非线性抛物型偏微分方程解的渐近性。关于不可压缩流的随机扰动的几个问题也将被研究。这些包括贝纳德对流的输运性质，随机扰动不可压缩流的大偏差，以及布朗电机的数学分析。提出的研究将为自然科学中积极讨论的几个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，这是研究各种种群进化的核心，如细菌，癌细胞，特定基因的携带者等，其中种群的每个成员可能会死亡或独立于其他成员产生后代。我们计划描述种群在不同空间区域的长期行为，除了分支，种群成员在空间中移动，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结合而引起的粒子（例如分子）的运动。特别是，我们将研究的机制，创造定向运动的波动随机或周期性的速度场，甚至在情况下，当场本身没有首选的方向。这种机制在许多应用中非常重要，并且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水平上。

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.