Boundary Behavior for Sticky Brownian Motion

Abstract

We study a reflecting Brownian motion on state space [0, ∞) by viewing it as a diffusion process associatedwith the infinitesimal generator A given by (Af)(x) = 1 2 f ′′(x) (0.1) and equipped with the boundary condition m({0}) 2 f ′′(0) = f ′ (0+). (0.2) This condition is referred to as a sticky boundary or a slowly reflecting boundary and we will call m({0}) the stickiness parameter. We will show, by means of a local time process, that the time spent at the boundary is in fact positive for such a process. We introduce the singular measure, the so called speed measure m, in order to investigate the occupation time at the boundary. Moreover, we will show that the sticky boundary condition (0.2) arises as a domain condition for the infinitesimal generator A of the reflecting Brownian motion.