  The Feynman-Kac formula implies that every suitable classical solution of a
semilinear Kolmogorov partial differential equation (PDE) is also a solution of
a certain stochastic fixed point equation (SFPE). In this article we study such
and related SFPEs. In particular, the main result of this work proves existence
of unique solutions of certain SFPEs in a general setting. As an application of
this main result we establish the existence of unique solutions of SFPEs
associated with semilinear Kolmogorov PDEs with Lipschitz continuous
nonlinearities even in the case where the associated semilinear Kolmogorov PDE
does not possess a classical solution.
