Random evolution theory provides models for systems where the mode of evolution is modified according to random changes in the environment. It was initiated at the end of the sixties by Hersh, Griego, Pinsky and other authors mainly in the case of linear diffusion–type evolution operators. Relevant results have been obtained especially in the analysis of related asymptotic problems. We propose an extension of the theory in the case of non linear evolution driven by Hamiltonians with switchings governed by a Markov chain. We put it in relation with a random Lax–Oleinik semigroup for which we prove continuity of the value function and existence of minimizing curves satisfying some differential equations. We apply the results to the analysis of an associated time–dependent system of weakly coupled Hamilton–Jacobi equations.