Stability of Constrained Markov‐Modulated Diffusions

A family of constrained diffusions in a random environment is considered. Constraint set is a polyhedral cone and coefficients of the diffusion are governed by, in addition to the system state, a finite‐state Markov process that is independent of the driving noise. Such models arise as limit objects in the heavy traffic analysis of generalized Jackson networks (GJN) with Markov‐modulated arrival and processing rates. We give sufficient conditions (which, in particular, includes a requirement on the regularity of the underlying Skorohod map) for positive recurrence and geometric ergodicity. When the coefficients only depend on the modulating Markov process (i.e., they are independent of the system state), a complete characterization for stability and transience is provided. The case, where the pathwise Skorohod problem is not well posed but the underlying reflection matrix is completely‐S, is treated as well. As consequences of geometric ergodicity various results, such as exponential integrability of invariant measures and central limit results (CLT) for fluctuations of long‐time averages of process functionals about their stationary values, are obtained. Conditions for stability are formulated in terms of the averaged drift, where the average is taken with respect to the stationary distribution of the modulating Markov process. Finally, steady‐state distributions of the underlying GJN are considered and it is shown that under suitable conditions, such distributions converge to the unique stationary distribution of the constrained random environment diffusion.