Geometric Ergodicity for Piecewise Contracting Processes with Applications for Tropical Stochastic Lattice Models

Stochastic lattice models are increasingly prominent as a way to capture highly intermittent unresolved features of moist tropical convection in climate science and as continuum mesoscopic models in material science. Stochastic lattice models consist of suitably discretized continuum partial differential equations interacting with Markov jump processes at each lattice site with transition rates depending on the local value of the continuum equation; they are a special case of piecewise deterministic Markov processes but often have an infinite state space and unbounded transition rates. Here a general theorem on geometric ergodicity for piecewise deterministic contracting processes is developed with full generality to apply to stochastic lattice models. A highly nontrivial application to the stochastic skeleton model for the Madden-Julian oscillation (Thual et al., 2013) is developed here where there is an infinite state space with unbounded and also degenerate transition rates. Geometric ergodicity for the stochastic skeleton model guarantees exponential convergence to a unique invariant measure that defines the statistical tropical climate of the model. Another application of the general framework is developed here for stochastic lattice models designed to capture intermittent fluctuation in the simplest tropical climate models. Other straightforward applications to models motivated by material science are mentioned briefly here.