Analytic and Probabilistic Krein-Rutman theorem for the reflected Brownian motion. Existence, uniqueness, and regularity of the quasistationary distribution.

par Loïs Delande (Cergy)

jeudi 19 nov. 2026, 11:15 → 12:15 Europe/Paris

Salle J. Cavailles (1R2-132)

In this talk, we investigate the reflected, drifted Brownian motion on a bounded domain. More precisely, we consider mixed absorbing-reflecting boundary conditions and investigate the notion of quasi-stationary distributions (QSD). We address two problems, which are solved completely independently of each other. First, using purely probabilistic tools, we prove existence and uniqueness of a QSD under minimal regularity assumptions on the coefficients. Second, using spectral theory and semigroup theory on Lp spaces, we study the analytical properties of the infinitesimal generator of the process we are interested in. It is a second order elliptic (non-symmetric) operator with mixed Dirichlet-Neumann boundary conditions. We establish regularity of its eigenfunctions up to the interface between the two boundary parts. Finally, using a Feynman-Kac formula, we are able to make a link between the two conducted analysis deriving in particular regularity for the QSD, but also characterization of the law of the exit point. Throughout, we work in domains in which the Dirichlet-Neumann junction is flat, we require no angle condition.