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Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. (English) Zbl 1365.60072
Summary: Brownian motion in \(\mathbb{R}_{+}^{2}\) 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 \(\mathbb{R}_{+}^{2}\) is found and its main term is identified depending on parameters \((\Sigma,\mu,R)\). For this purpose the analytic approach of G. Fayolle et al. [Random walks in the quarter-plane. Algebraic methods, boundary value problems and applications. Berlin: Springer (1999; Zbl 0932.60002)] and V. A. Malyshev [Sib. Math. J. 14, 109–118 (1973; Zbl 0307.60060); translation from Sib. Mat. Zh. 14, 156–169 (1973)], restricted essentially up to now to discrete random walks in \(\mathbb{Z}_{+}^{2}\) with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on \(\mathbb{R}_{+}^{2}\) with reflections on the axes.

MSC:
60J65 Brownian motion
05A15 Exact enumeration problems, generating functions
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
60K25 Queueing theory (aspects of probability theory)
30F10 Compact Riemann surfaces and uniformization
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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