Exit-time of an inhomogeneous diffusion

We investigate the exit problem for a diffusion which drift is not time-homogeneous. More precisely, we study this problem for a McKean-Vlasov diffusion, that corresponds to the probabilistic interpretation of the granular media equation. This problem has already been solved in previous articles when the confining potential is uniformly strictly convex. Two different methods have been used. However, these two methods do not extend to the non-convex case. Consequently, here, we proceed in another way: by making a coupling with another McKean-Vlasov diffusion with a uniformly strictly convex confining potential. We present the result in a simple case, the one in which the interacting potential is linear. However, the result can be extended in a more general setting.