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

Abstract

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.

Integral expression for the stationary distribution of reflected Brownian motion in a wedge

“…This process has been extensively explored, and its multidimensional version (a semimartingale reflected Brownian motion in the positive orthant R d + , as well as in convex polyhedrons) as well. It has been studied in depth, with focuses on its definition and semimartingale properties [54,55,16,57], its recurrence or transience [55,13,35,9,7,6,15], the possible particular (e.g., product) form of its stationary distribution [34,19,46], its Lyapunov functions [23], its links with other stochastic processes [41,22,42], its use to approximate large queuing networks [26,2,33,38,37], the asymptotics of its stationary distribution [30,17,28,48], numerical methods to compute the stationary distribution [13,14], links with complex analysis [26,2,8,29], comparison and coupling techniques [50,49], etc.…”

“…See Remarks 7 and 18 for further related comments. • The paper [28] obtains the exact asymptotics of the stationary distribution along any direction in the quarter plane, see [28,. Constants in these asymptotics involve the functions ϕ 1 and ϕ 2 in (4), and can thus be made explicit with Theorem 1.…”

“…The domain in (15) strictly contains G R , see [28]. Thanks to Lemma 3, this implies that the Laplace transform ϕ 1 (θ 2 ) can be extended meromorphically to a domain containing G R .…”

“…The domain (15) is simply connected by [28, §2.4]. The formula (16), see [28,Lemma 6], is a direct consequence of the functional equation (5) evaluated at (Θ − 1 (θ 2 ), θ 2 ), first on the (non-empty) open domain…”

“…Note, this is peculiar to the case of a drift with negative coordinates (our hypothesis (13)). More details can be found in the proof of Theorem 11 in [28].…”