Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

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.