Moment bounds and geometric ergodicity of diffusions with random switching and unbounded transition rates

Abstract

A diffusion with random switching is a Markov process that consists of a stochastic differential equation part X t and a continuous Markov jump process part Y t . Such systems have a wide range of applications, where the transition rates of Y t may not be bounded or Lipschitz. A new analytical framework is developed to understand the stability and ergodicity of these processes and allows for genuinely unbounded transition rates. Assuming the averaged dynamics is dissipative, the first part of this paper explicitly demonstrates how to construct a polynomial Lyapunov function and furthermore moment bounds. When the transition rates have multiple scales, this construction comes interestingly as a dual process of the averaging of fast transitions. The coefficients of the Lyapunov function can be seen as the potential dissipation of each regime in different scales, and a comparison principle comes naturally under this interpretation. On the basis of these results, the second part of this paper establishes geometric ergodicity for the joint processes. This can be achieved in two scenarios. If there is a commonly accessible regime that satisfies the minorization condition, the geometric convergence to the ergodic measure takes place in the total variation distance. If there is contraction on average, the geometric convergence takes place in a proper Wasserstein distance and is proved through an application of the asymptotic coupling framework. BackgroundDiffusions with random switching are stochastic processes consisting of two components: a diffusion process X t in R d and a continuous jump process Y t on a finite set F . The dynamics of X t follows a stochastic differential equation (SDE)(1.1) Throughout, we assume b(x, y) is C 1+δ and σ (x, y) is C 2+δ in x for some δ > 0. The behavior of Y t can be described by a transition rate function λ (x, y,ỹ), in other wordsWe denote the total transition rate asλ(x, y) = ỹ =y λ (x, y,ỹ), and the joint process as Z t = (X t , Y t ), which takes place in the space E = R d × F .

On subgeometric ergodicity of regime-switching diffusion processes

“…This shows that the behavior of these type of processes depends on an (sometime very nontrivial) interplay between both (continuous-state and regime-switching) components. By using the classical Foster-Lyapunov method for geometric ergodicity of Markov processes, in [1], [2], [11], [13], [16], [22], [20] , [23], [26], [28] and [29] geometric ergodicity with respect to the total variation distance of regime-switching diffusions is established. In this article, we employ the Foster-Lyapunov method for subgeometric ergodicity of Markov processes developed in [6] and obtain conditions ensuring subgeometric ergodicity of this class of processes.…”

“…For Markov processes that do not converge in total variation, ergodic properties under Wasserstein distances are studied since this distance function, in a certain sense, induces a finner topology (see [18] and [25]). In [2], [21] and [23] the coupling approach together with the Foster-Lyapunov method is employed to establish geometric contractivity and ergodicity of the semigroup of a regime-switching diffusion with respect to a Wasserstein distance. In this article, by using the ideas developed in [12], in the context of diffusion processes, and combining the asymptotic flatness conditions in eqs.…”

Ergodicity for neutral type SDEs with infinite length of memory

“…In case the noise is path‐dependent, we would like to emphasize that the ergodicity under the total variational distance and the strong Feller property are not available since the laws of functional solutions with different initial data are mutually singular. The weak Harris' theorem has been applied in, e.g., [7, 11, 23, 29] to establish the ergodicity for highly degenerate stochastic dynamical systems including Markov processes with random switching.…”

On the monotonicity property of the generalized eigenvalue for weakly-coupled cooperative elliptic systems

“…In fact, finding sufficient condition for the existence of Lyapunov function is an important issue. See, for instance, [9,[42][43][44][45] and references therein. In particular, by [44,Theorem 5.3], if V is inf-compact, M is bounded and a is uniformly elliptic, we get V-geometric ergodicity for the regime switching diffusion.…”