Martin Boundary of a Degenerate Reflected Brownian Motion in a Wedge

We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green’s functions associated with the process. These Laplace transforms are expressed as an infinite sum of products by iterating a functional equation, which is deeply linked to the compensation method. We also derive the asymptotics of the Green’s functions along all possible paths and determine the (minimal) Martin boundary. Finally, we provide explicit formulae for all the corresponding positive harmonic functions. Funding: This project received funding from ENS Rennes (PhD funding) and from the ANR RESYST [ANR-22-CE40-0002].