# Ergodic Backward SDEs: Multiplicative and Degenerate Noise Case

We present here two different ‘non standard’ cases in which Ergodic BSDEs can be applied, in an infinite dimensional framework, to represent the solution to a nonlinear Poisson PDE together with the value function of a related Ergodic optimal control problem. In a first case, we allow multiplicative noise (and consequently non constant second order operator in the PDE) assuming convexity in the gradient of the nonlinear term. In the second, the noise is additive but degenerate and we drop the so called ‘structure’ condition’. On this last situation the problem is treated by the ‘randomization’ technique.
This is a joint work with Giuseppina Guatteri (Politecnico di Milano) and Andrea Cosso (Università di Bologna).