Entropic metastability for drifted reflected Brownian motion: quasi-stationary distributions, regularity, and the exit position distribution

Source abstract

Motivated by entropic metastability and the narrow escape problem, we investigate quasi-stationary distributions for reflected, drifted elliptic diffusions under mixed absorbing-reflecting boundary conditions on a bounded domain. We address two problems, which are solved completely independently of each other. First, using purely probabilistic tools, we prove the existence and uniqueness of a quasi-stationary distribution under minimal regularity assumptions on the coefficients (the case of an unbounded domain is also analyzed). Second, using spectral theory and the theory of strongly continuous semigroups on L p spaces, we study the analytical properties of the quasistationary distribution. We relate it to the principal eigenfunction of the adjoint of the associated (non-symmetric) infinitesimal generator with mixed absorbing-reflecting boundary conditions, and we establish its regularity up to the interface between the two boundary parts. We also fully characterize the law of the exit point when the process is initially distributed according to its quasistationary distribution. In particular, we work on domains in which the absorbing (Dirichlet) and reflecting (Neumann) parts of the boundary meet tangentially, without forming any non-trivial angle, a configuration that raises genuine regularity difficulties for the solutions of the associated mixed boundary value problem.