Asympotic Problems in Random Transport

本提案的主要目标是研究输运现象的各个方面。一类问题涉及粒子在随机力场中的运动。我们将证明，当时间趋于无穷时，粒子的位置、速度和能量，在适当的尺度下，收敛于某些极限过程。将根据力场的分布明确地描述极限过程。我们还将研究随机速度场携带的测度，并描述它们在时间趋于无穷时的极限行为。另一组问题涉及哈密顿流的随机扰动。我们将努力获得紧辛流形上随机摄动哈密顿流的有效行为的描述。该提案的一个单独部分致力于吉布斯场的反问题。我们将证明总是存在具有规定的相关函数的吉布斯场，只要相关函数满足一定的一致性条件。大多数提出的问题自然产生于对物理学、海洋学、湍流理论和统计力学等各种现象的研究中。例如，考虑一个粒子在力场中运动，例如一个带电粒子在电磁场中运动。我们打算根据场的概率性质来描述粒子的行为（如它的位置、速度和能量）。主要的假设是，粒子被观察到的时间比力场的典型尺寸大得多。我们还将研究随机流的输运性质。人们可以把大气气流携带的污染物看作是这类问题的物理模型。根据流动的概率性质，人们希望描述它所携带的物质的长时间行为。几组研究人员在各种简化假设下，如流动的周期性，已经在这个方向上得到了许多结果。我们将研究一大类物理相关流的问题。

The main goal of this proposal is to investigate various aspects of transport phenomena. One type of problems concerns the motion of a particle in a random force field. We shall prove that the position, velocity, and energy of the particle, when scaled appropriately, converge, as time tends to infinity, to certain limiting processes. The limiting processes will be described explicitly in terms of the distribution of the force field. We shall also study measures carried by random velocity fields and describe their limiting behavior as time tends to infinity. Another set of problems concerns random perturbations of Hamiltonian flows. We shall strive to obtain a description of the effective behavior of randomly perturbed Hamiltonian flows on compact symplectic manifolds. A separate part of the proposal is devoted to an inverse problem for Gibbs fields. We shall prove that there always exist Gibbs fields with prescribed correlation functions, provided that the correlation functions satisfy certain consistency conditions. Most of the proposed problems naturally arise in the study of various phenomena in physics, oceanography, theory of turbulence, and statistical mechanics. Consider, for example, a particle moving in a force field, such as a charged particle in an electro-magnetic field. We intend to describe the behavior of the particle (such as its position, velocity and energy) based on the probabilistic properties of the field. The main assumption is that the time at which the particle is observed is much larger than the typical size of the force field. We shall also study the transport properties of random flows. One could think of a pollutant carried by atmospheric currents as a physical model for this type of problems. Based on the probabilistic properties of the flow, one wishes to describe the long time behavior of the substance carried by it. Many results in this direction have been obtained be several groups of researchers under various simplifying assumptions, such as the periodicity of the flow. We shall study the problem for a large class of physically relevant flows.