论文标题
通过功能随机微分方程的随机涡流动力学
Random vortex dynamics via functional stochastic differential equations
论文作者
论文摘要
在本文中，我们介绍了一个新颖的，封闭的三维（3D）随机涡流动力学系统，该系统等效于Navier-用于不可压缩的粘性流体流动方程。新的随机涡流动力学系统由随机微分方程组成，与二维随机涡流动力学方程相反，该方程与有限的普通功能差分方程相反。这种新的随机涡流系统为设计新的数值方案（随机涡流方法）铺平了道路，用于通过蒙特卡洛模拟解决三维不可压缩流体流动方程。为了得出3D随机涡流动力学方程，我们开发了两个功能强大的工具：第一个是几个泰勒扩散的条件分布的双重性，该分布提供了按零件进行集成的路径空间版本；第二个是远期类型的Feynman-kac公式，它代表了功能积分的非线性抛物线方程的解决方案。这些技术工具和潜在的想法可能对处理其他非线性问题有用。
In this paper we present a novel, closed three-dimensional (3D) random vortex dynamics system, which is equivalent to the Navier--Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled with a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the 3D random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman--Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.
