In this talk, we deal with Ergodic Backward Stochastic Differential Equations for which the
underlying SDE (in finite dimension) has a sublinear diffusion. Hu, Madec and Richou have recently studied those
equations, but with an additive noise (and in infinite dimension). First, we show existence and
uniqueness of the solution under assumptions similar to their work (especially weak dissipativity of the drift for the underlying SDE). Then, we obtain
the large time behaviour of viscosity solutions of HJB equations.