Abstract
Brownian motion in R 2 + with covariance matrix $\Sigma$ 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 ($\Sigma$, $\mu$, 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.
Highlights
Our aim is to obtain the asymptotic expansion of the stationary distribution density π(x) = π(x1, x2) as x1, x2 → ∞ and x2/x1 → tan (α0) for any given angle α0 ∈ [0, π/2]
We present at the same time the organization of the article
We would like to compute their asymptotic expansion as r → ∞ and prove it to be uniform for α fixed in a small neighborhood O(α0), α0 ∈]0, π/2[
Summary
Two-dimensional semimartingale reflecting Brownian motion (SRBM) in the quarter plane received a lot of attention from the mathematical community. Problems such as SRBM existence [39, 40], stationary distribution conditions [19, 22], explicit forms of stationary distribution in special cases [7, 8, 19, 23, 30], large deviations [1, 7, 33, 34] construction of Lyapunov functions [10], and queueing networks approximations [19, 21, 31, 32, 43] have been intensively studied in the literature. Many results on two-dimensional SRBM have been fully or partially generalized to higher dimensions
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