Ergodic distribution dependent BSDE and application to long-time behavior of finite horizon distribution dependent BSDE

After proving existence and uniqueness of ergodic distribution dependent backward stochastic differential equations (BSDEs) under strong and weak dissipativity regimes for the underlying McKean--Vlasov SDE, we leverage this new framework to investigate the long-time behavior of distribution dependent BSDEs on a finite-time horizon. Finally, we apply our results to solve an ergodic McKean--Vlasov stochastic control problem and study the long-time behavior of the value function of a finite-horizon McKean--Vlasov stochastic control problem.