论文标题
在非线性feynman-kac公式上，用于半线性抛物线偏微分方程的粘度解决方案
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
论文作者
论文摘要
经典的Feynman-KAC身份通过为线性kolmogorov PDE的经典解提供随机分析方程（PDE）之间建立一个随机分析和部分微分方程（PDE）的桥梁。这为基于采样的蒙特卡洛近似方法的推导打开了大门，该方法可以无网格，从而有机会近似PDE的解决方案而不会受到维数的诅咒。在本文中，我们将经典的Feynman-kac公式扩展到某些半线性Kolmogorov PDE。更具体地说，我们确定了随机固定点方程（SFPE）的合适溶液，当将经典的Feynman-KAC身份正式应用于半线性Kolmorogov PDES时，它会成为相应PDES的粘度解决方案。这尤其证明了使用全历史的递归多级PICARD（MLP）近似算法的合理性，最近已证明在SFPE的溶液的数值近似中，在SFPE的溶液的数值近似中，在半线性Kolmogorov PDES的数值近似中都可以克服尺寸的诅咒。
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman-Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs.
