Mathematical Sciences: Stochastic Partial Differential Equations -- A Particle Systems Approach

研究者正在发展一种随机偏微分方程（SPDEs）的微积分，用于粒子和漩涡的分布。这些偏微分方程是抛物线型偏微分方程，通过附加的波动项进行校正。涨落项必须由粒子分布导出。这种方法的优点之一是，SPDE中的所有量都可以用物理术语来解释，如果SPDE中的守恒定律适用于底层物理模型，那么它们将适用于SPDE。研究者正在开发一种研究随机系统的方法，这些系统出现在理论物理、流体动力学、化学和数学的各个领域。模型中的数量可以用物理术语来解释，这应该有助于分析。

The investigator is developing a calculus of stochastic partial differential equations (SPDEs) for the distribution of particles and vortices. These SPDEs are partial differential equations (PDEs) of parabolic type which are corrected by an additional fluctuation term. The fluctuation term has to be derived from the particle distribution. One of the advantages of this approach is that all quantities in the SPDE can be interpreted in physical terms, and conservation laws will hold for the SPDE if they hold for the underlying physical model. The investigator is developing a method of studying stochastic systems which arise in theoretical physics, fluid dynamics, chemistry and various fields of mathematics. The quantities in the models can be interpreted in physical terms, which should help in the analysis.