Quantitative ergodicity for some switched dynamical systems

Benaı̈m, Michel; Borgne, Stéphane Le; Malrieu, Florent; Zitt, Pierre-André

doi:10.1214/ecp.v17-1932

Cited by 62 publications

(99 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A more suitable distance for our purpose is the Wasserstein distance, which is also used in previous works for models with bounded or Lipschitz transition [3,4,7,36]. For any distance d on a space E, the associated Wasserstein distance between two measures μ, ν on E is defined as:…”

Section: Wasserstein Metric Convergence With Contraction On Averagementioning

confidence: 99%

“…In this section we will discuss the consequence of this assumption and also provide a simple way to verify in Lemma 4.2. Most derivations here are relatively standard and may have a simpler version in [4,5,7] when the transition rates are bounded. We provide the complete proofs here to be self-contained.…”

Section: Accessibility Analysismentioning

confidence: 99%

“…It can be seen as the Y t part in our joint process. 4. In filtering and predictive modeling [16,30,32], diffusions with random switching are used as test beds to quantify the uncertainty from model errors.…”

Section: Introductionmentioning

confidence: 99%

“…In the last decade, a series of works have been devoted for the questions above [1,[3][4][5]7,11,39,46]. In the simplest setting, the transition rates are constants, λ(x, y, y ) = λ(y, y ), and X t is driven by the linear equation (1.2).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Tong

Majda

2016

Res Math Sci

View full text Add to dashboard Cite

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 .

show abstract

Section: Wasserstein Metric Convergence With Contraction On Averagementioning

confidence: 99%

Section: Accessibility Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Tong

Majda

2016

Res Math Sci

View full text Add to dashboard Cite

show abstract

“…Recently, the asymptotic behavior of PDMPs have been investigated in [4,5,12,37], limit theorems for infinite dimensional PDMPs in [35], control problems in [10,11,22], numerical methods in [33,6], time reversal in [27] and to end up this list with no claim of completeness, estimation of the jump rates for PDMPs in [2]. Hybrid systems are the object of great attention because they offer an accurate description of a large class of phenomena arising in various domains such as physics or biology.…”

Section: Introductionmentioning

confidence: 99%

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

Genadot

Thieullen

2014

ESAIM: PS

View full text Add to dashboard Cite

In [20], the authors addressed the question of the averaging of a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimension. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuation of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated Langevin approximation is considered. The motivation of this work is a stochastic Hodgkin-Huxley model which describes the propagation of an action potential along the nerve fiber. We study this PDMP in detail and provide more general results for a class of Hilbert space valued PDMP.

show abstract

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

Majda

Tong

2015

Comm Pure Appl Math

View full text Add to dashboard Cite

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.

show abstract

Quantitative ergodicity for some switched dynamical systems

Cited by 62 publications

References 39 publications

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

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

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

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

Contact Info

Product

Resources

About