Thermodynamic formalism for random countable Markov shifts

We introduce a relative Gurevich pressure for random countable topologically mixing Markov shifts. It is shown that the relative variational principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous functions. This is accomplished via a new construction of an equivariant family of fiber measures using Crauel's relative Prohorov theorem. Some properties of the Gibbs measures are discussed as well.