Ergodic BSDEs with Multiplicative and Degenerate Noise

Abstract: In this paper we study an Ergodic Markovian BSDE involving a forward process $X$ that solves an infinite dimensional forward stochastic evolution equation with multiplicative and possibly degenerate diffusion coefficient. A concavity assumption on the driver allows us to avoid the typical quantitative conditions relating the dissipativity of the forward equation and the Lipschitz constant of the driver. Although the degeneracy of the noise has to be of a suitable type we can give a stochastic representation of a large class of Ergodic HJB equations; morever our general results can be applied to get the synthesis of the optimal feedback law in relevant examples of ergodic control problems for SPDEs.