Mathematical Sciences: Reciprocal Diffusions, Stochastic Differential Equations of Second Order and Conservation Laws

本课题主要研究互易扩散、二阶随机微分方程组和非线性守恒律组。首席研究员最近发展了一种二阶互易扩散过程理论。这推广了著名的马尔可夫扩散的一阶理论。一阶和二阶是关于时差和区分的。首席调查者已经证明，倒数扩散过程满足适当定义的二阶随机微分方程。密度、平均动量和平均动量通量满足一组类似于流体力学的非线性守恒定律。将探索这些思想在随机积分、流体力学、随机最优控制和量子力学中的应用。

This project concerns the study of reciprocal diffusions, second order stochastic differential equations and system of nonlinear conservation laws. The Principal Investigator has recently developed a theory of second order, reciprocal diffusion processes. This generalizes the well known first order theory of Markov diffusion. First and second order are with respect to time differences and differentiations. The Principal Investigator has shown that a reciprocal diffusion process satisfies a stochastic differential equation of second order appropriately defined. The density, mean momentum and mean momentum flux satisfy a system of nonlinear conservation laws similar to those of fluid mechanics. Applications of these ideas in stochastic integration, fluid mechanics, stochastic optimal control, and quantum mechanics will be explored.