Recently, there has been increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this talk we will present some results on the contact process in a evolving (edge) random environment on (infinite) connected and transitive graphs with bounded degrees. We assume that the evolving random environment is described by an autonomous ergodic spin systems with finite range, for example by dynamical percolation. This background process determines which edges are open or closed for infections.

Most contribution to this class of models are very recent and we will give a short overview. Our own resulsts concern the dependence of the critical infection rate for survival on the random environment and on the initial configuration of the system. For the latter we state sufficient conditions such that the initial configuration of the system has no influence on the phase transition between extinction and survival. We show that this phase transition coincides with the phase transition between a trivial/non-trivial upper invariant law. We also discuss the continuity properties of the survival probability as well as conditions for complete convergence.
Finally, we touch upon considering the contact process on dynamical long range percolation and discuss some open problems, conjecture and further research directions.

This is joint work with Marco Seiler (University of Göttingen).