Autostabilizing nonlinear reflected process (Q1272906)

Consider the reflected nonlinear stochastic differential equation: \[ dX_t=dB_t-b* u_t(X_t) dt-{\mathbf{1}}_{\{-1,1\}} (X_t)dk_t,\;X_t\in[-1,1],\;\forall t\geq 0,\quad u_t(dx)= \mathbb{P}(X_t\in dx). \] Existence and uniqueness of the solution \((X_t,k_t,u_t)\) of such equations (in \(d\) dimensions) have been proved by Sznitman. Under some hypothesis on the function \(b\) (odd, increasing, Lipschitz, and a technical one) the authors prove that the process \(X_t\) has a unique invariant measure (with an even density), and that the law of \(X_t\) converges towards this measure.