Subexponential rate of convergence of a class of Levy-driven SDEs and applications in stochastic networks

       We study ergodic properties of a class of multidimensional piecewise OU processes with jumps. They include the scaling limits arising from stochastic networks with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime. These models have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Levy process, or (2) an anisotropic Levy process with independent 1-dim symmetric α-stable components, or (3) an anisotropic Levy process as in (2) and a pure-jump Levy process. We also study the class of models driven by a subordinate Brownian motion. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity. These assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. For stochastic network models, the rate of convergence is polynomial and provide a sharp quantitative characterization of the rate via matching upper and lower bounds.