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The present paper is devoted to the study of the asymptotic behavior of thevalue functions of both finite and infinite horizon stochastic control problemsand to the investigation of their relation with suitable stochastic ergodiccontrol problems. Our methodology is based only on probabilistic techniques, asfor instance the so-called randomization of the control method, thus avoidingcompletely analytical tools from the theory of viscosity solutions. We are thenable to treat with the case where the state process takes values in a general(possibly infinite dimensional) real separable Hilbert space and the diffusioncoefficient is allowed to be degenerate.

Ergodic Control of Infinite Dimensional SDEs with Degenerate Noise