An ergodic backward stochastic differential equation approach to large time behaviour of some parabolic semilinear PDEs

In this talk we study the large time behaviour of some semilinear parabolic PDEs by a purely probabilistic approach. For that purpose, we show that the solution of a backward stochastic differential equation (BSDE) in finite horizon $T$ taken at initial time behaves like a linear term in $T$ shifted with a solution of the associated ergodic BSDE taken at inital time. Moreover we give an explicit rate of convergence: we show that the following term in the asymptotic expansion has an exponential decay. This is a Joint work with Ying Hu and Pierre-Yves Meyer from Rennes (IRMAR - France).