Asympotic Problems in Random Transport

这项提议的主要目标是调查运输现象的各个方面。一类问题涉及粒子在随机力场中的运动。我们将证明，粒子的位置、速度和能量，在适当的尺度下，随着时间趋于无穷大，会收敛到某些极限过程。极限过程将根据力场的分布来明确地描述。我们还将研究随机速度场所承载的测量，并描述它们在时间趋于无穷大时的极限行为。另一组问题涉及哈密顿流的随机扰动。我们将努力得到紧辛流形上随机扰动哈密顿流的有效行为的描述。该提案的另一部分致力于吉布斯场的反问题。我们将证明，如果关联函数满足一定的一致性条件，则总是存在具有指定关联函数的Gibbs场。大多数提出的问题自然会出现在物理、海洋学、湍流理论和统计力学中的各种现象的研究中。例如，考虑在力场中运动的粒子，例如在电磁场中的带电粒子。我们打算基于场的概率属性来描述粒子的行为(如其位置、速度和能量)。主要的假设是，观察到粒子的时间比力场的典型大小大得多。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想象成这类问题的物理模型。基于流动的概率性质，人们希望描述它所携带的物质的长期行为。在各种简化假设下，例如流动的周期性，几组研究人员已经在这个方向上取得了许多结果。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.