We consider a general class of random evolutions which we call semilinear. We establish and discuss equations from which the first two moments of such processes can be computed. This is done by proving a generalization of the Feynman–Kac formula and using a duality argument between a backward and a forward equation. We thus extend some results (first established by Frisch in Probabilistic Methods in Applied Mathematics, A. T. B. Reid, ed., Academic Press, New York, 1968), which are known for stochastic differential equations with Markovian coefficients, to stochastic partial differential equations and also to some non–Markovian cases, see examples in §§ 5 and 6.