Abstract: We study the ergodic properties of a class of multidimensional piecewise Ornstein--Uhlenbeck

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 as special cases.

In these models, the Ito equations 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 one-dimensional symmetric

\alpha-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, which contains an isotropic (or rotationally

invariant) \alpha-stable Levy process as a special case. We identify conditions on the parameters in the drift,

the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity.

We show that these assumptions are sharp, and we identify some key necessary conditions for the process to

be ergodic. For the stochastic network models, we show that the rate of convergence is polynomial and provide a sharp quantitative characterization of the rate via matching upper and lower bounds.