Contact process on a dynamical long range percolation

In this paper we introduce a contact process on a dynamical long range percolation (CPDLP) defined on a complete graph $(V,\mathcal{E})$. A dynamical long range percolation is a Feller process defined on the edge set $\mathcal{E}$, which assigns to each edge the state of being open or closed independently. The state of an edge $e$ is updated at rate $v_e$ and is open after the update with probability $p_e$ and closed otherwise. The contact process is then defined on top of this evolving random environment using only open edges for infection while recovery is independent of the background. First, we conclude that an upper invariant law exists and that the phase transitions of survival and non-triviality of the upper invariant coincide. We then formulate a comparison with a contact process with a specific infection kernel which acts as a lower bound. Thus, we obtain an upper bound for the critical infection rate. We also show that if the probability that an edge is open is low for all edges then the CPDLP enters an immunization phase, i.e. it will not survive regardless of the value of the infection rate. Furthermore, we show that on $V=\mathbb{Z}$ and under suitable conditions on the rates of the dynamical long range percolation the CPDLP will almost surely die out if the update speed converges to zero for any given infection rate $\lambda$.