Fractional Boundary Value Problems and elastic sticky Brownian motions, II: Non-local dynamic boundary conditions on smooth domains

Abstract

We introduce a wide class of sticky Brownian motions (not necessarily Markovian) and study their boundary occupation time through different notions of holding times. This class consists of all processes obtained from various symbols of subordinators, say $Φ$. Our discussion focuses on the infinite activity case and, as a reference model, we consider the stable subordinator corresponding to $Φ(λ)=λ^α$. Sticky diffusion processes on bounded domains can spend a finite time (with a finite mean) on the lower-dimensional space given by the boundary. Once the process hits the boundary, it restarts either instantaneously or after a random period of time. While on the boundary, it can stay or move according to dynamics that differ from those in the interior. Such processes may be characterized by a time derivative appearing in the boundary condition of the governing problem. We restrict our attention to static behaviour without lateral diffusion (i.e., the boundary trace process is a pure jump process). We use suitable time-changes in order to describe fractional sticky conditions and the associated boundary behaviours. We show that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions, which are strong Markov in the interior and spend a finite time (with an infinite mean) on the boundary. Such behaviour can be interpreted as a trapping effect from a macroscopic point of view.