On the most general one-dimensional Brownian Motion with boundary conditions at the origin.

Abstract

In this master’s dissertation, we characterize the most general one-dimensional Brownian motion under some Markovian behavior at zero via the study of its infinitesimal generators. The class of processes here considered is defined as the class of diffusion processes that behave as the absorbed Brownian Motion up to the hitting time of zero, and at zero the process has some (Markovian) behavior, which includes jumping to an extra absorbing point ∆ called cemetery. Carefully adapting techniques of Knight’s book [3], we obtain two new results. Our first main result consists on proving that the most general Brownian motion on the state space R∪{∆} coincides with the Skew Sticky Killed Brownian Motion, whose infinitesimal generator can be found in Borodin’s book [1]. Our second main result consists on the characterization of the most general Brownian motion on the state space (−∞,0−] ∪ [0+,∞) ∪ {∆}. We conclude that the class of processes obtained includes, as a particular case, the Snapping Out Brownian Motion, a Brownian motion on (−∞,0−]∪[0+,∞) recently constructed in Lejay’s paper [5]. Moreover, the class of processes here obtained includes a Brownian-type process not known in the literature, which we call a Skew Sticky Killed Snapping Out Brownian motion.