Exponential ergodicity for Markov processes with random switching
TL;DR: In this article, the convergence to equilibrium in terms of Wasserstein distance has been studied for piecewise deterministic Markov processes with two components, where the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component.
Abstract: We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.
Citations
More filters
TL;DR: In this paper, a non-autonomous ODE on a smooth manifold with right-hand side that randomly switches between the elements of a finite family of smooth vector fields is considered, and it is shown that Hormander type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure.
Abstract: We consider a non-autonomous ordinary differential equation on a smooth manifold, with right-hand side that randomly switches between the elements of a finite family of smooth vector fields. For the resulting random dynamical system, we show that Hormander type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure.
71 citations
TL;DR: The existence and pathwise uniqueness of regime-switching diffusion processes in an infinite state space are established and the strong Feller properties of these processes are investigated by using the theory of parabolic differential equations and dimensional-free Harnack inequalities.
Abstract: We establish the existence and pathwise uniqueness of regime-switching diffusion processes in an infinite state space, which could be time-inhomogeneous and state-dependent. Then the strong Feller properties of these processes are investigated by using the theory of parabolic differential equations and dimensional-free Harnack inequalities.
52 citations
Posted Content•
TL;DR: In this article, the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response was studied. And the authors used the ergodicity of regime switching diffusion processes to justify whether a predator die out or not when it will die out in some environments and not in others.
Abstract: In this work we study the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response. The switching process is used to describe the random changing of corresponding parameters such as birth and death rates of a species in different environments. Our criteria can justify whether a prey die out or not when it will die out in some environments and will not in others. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.
52 citations
TL;DR: More general stochastic hybrid systems are considered and explicit formulae for various statistics of the solution of the heat equation with randomly switching boundary conditions are found and almost sure results about its regularity and structure are obtained.
Abstract: We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.
52 citations
TL;DR: In this article, the authors provided some on-off type criteria for recurrence of regime switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem.
Abstract: We provide some on-off type criteria for recurrence of regime-switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and an infinite countable space are both studied. Especially, we put forward a finite partition method to deal with switching process in an infinite countable space. As an application, we study the recurrence of regime-switching Ornstein-Uhlenbeck process, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.
41 citations
References
More filters
Book•
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.
5,524 citations
TL;DR: In this paper, the authors developed criteria for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator, and applied the criteria to several specific processes, including linear stochastic systems under nonlinear feedback, work-modulated queues, general release storage processes and risk processes.
Abstract: In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.
1,000 citations
TL;DR: In this article, the authors define the Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are.
728 citations
Book•
01 Jan 1998
TL;DR: Survey of iterated random functions offers a method for studying the steady state distribution of a Markov chain, and presents useful bounds on rates of convergence in a variety of examples.
Abstract: Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average.
693 citations
TL;DR: In this article, various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) were presented.
Abstract: We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semigroup. © 2004 Wiley Periodicals, Inc.
517 citations