  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.
