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 paper, 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.

中文翻译：

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关于半线性抛物型偏微分方程粘度解的非线性费曼-卡克公式

经典的 Feynman-Kac 恒等式通过为线性 Kolmogorov PDE 的经典解提供随机表示，在随机分析和偏微分方程 (PDE) 之间架起了一座桥梁。这为基于采样的蒙特卡罗逼近方法的推导打开了大门，该方法可以是无网格的，从而有机会逼近 PDE 的解，而不会受到维数灾难的影响。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，我们确定了随机不动点方程 (SFPE) 的合适解，该解是在将经典 Feynman-Kac 恒等式正式应用于半线性 Kolmorogov PDE 时出现的，作为相应 PDE 的粘度解。这尤其证明了