Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach

Brownian motion in R 2 + with covariance matrix $\mathrm{\Sigma }$ and drift $\mu$ 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 ($\mathrm{\Sigma }$, $\mu$, 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.