On the Ergodic Properties and Invariant Measure of a Two-Dimensional Reflected Ornstein–Uhlenbeck Process

We consider a two-dimensional reflected Ornstein–Uhlenbeck (ROU) process that arises as the diffusion approximation for a parallel server network with a randomly split Hawkes arrival process (or a multivariate Hawkes arrival process) in heavy traffic. We study the ergodic properties of the process, including the positive recurrence and rate of convergence in total variation distance and in Wasserstein distance. We also provide a numerical scheme based on a Monte Carlo method to approximate the invariant measure of the process. Funding: G. Pang is partly supported by the National Science Foundation [Grants DMS 2216765 and CMMI 2452829].