Brownian motion in R 2 + with covariance matrix Σ and drift μ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters (Σ, μ, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on the axes.

We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp. non-rational), then the Martin boundary is discrete, homeomorphic to Z (resp. continuous, homeomorphic to R). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.

Brownian motion in R 2 + with covariance matrix Σ and drift μ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters (Σ, μ, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on the axes.