Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution

Champagnat, Nicolas; Villemonais, Denis

doi:10.30757/alea.v16-49

Cited by 7 publications

(10 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As most limit distributions cannot be given in explicit forms, it is natural to ask if an a priori estimate can established [13,27,35,50]. Our result answers this in the affirmative.…”

Section: Introductionmentioning

confidence: 51%

The asymptotic tails of limit distributions of continuous time Markov chains

Xu¹,

Hansen²,

Wiuf³

2020

Preprint

View full text Add to dashboard Cite

This paper investigates tail asymptotics of stationary distributions and quasistationary distributions of continuous-time Markov chains on a subset of the non-negative integers. A new identity for stationary measures is established. In particular, for continuoustime Markov chains with asymptotic power-law transition rates, tail asymptotics for stationary distributions are classified into three types by three easily computable parameters: (i) Conley-Maxwell-Poisson distributions (light-tailed), (ii) exponential-tailed distributions, and (iii) heavy-tailed distributions. Similar results are derived for quasi-stationary distributions. The approach to establish tail asymptotics is different from the classical semimartingale approach. We apply our results to biochemical reaction networks (modeled as continuous-time Markov chains), a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth-death processes.

show abstract

“…As most limit distributions cannot be given in explicit forms, it is natural to ask if an a priori estimate can established [13,27,35,50]. Our result answers this in the affirmative.…”

Section: Introductionmentioning

confidence: 51%

The asymptotic tails of limit distributions of continuous time Markov chains

Xu¹,

Hansen²,

Wiuf³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Besides, for another class of mean-field particle systems, the McKean-Vlasov diffusions (for which interaction is induced by an interaction potential force in the drift of the diffusion), it is well-known that there are cases (in non-convex confining potential with convex interaction for instance, at low temperature, for instance) where the non-linear limit system has several equilibria and the convergence rate of the particle system (which has a unique invariant measure for all N ) goes to +∞ with N . In fact we don't expect this to happen for the Fleming-Viot particle system in our context (compact space, elliptic diffusion, smooth killing) where the QSD is unique and the long-time convergence of the limit process and the uniform in time propagation of chaos have been established in non-perturbative cases (for a discussion on other cases where the QSD may not be unique, we refer to [1,11]). This may indicate that the uniform in N long-time convergence could hold in much more general cases, far from the perturbative regime around the non-interacting case.…”

Section: Related Workmentioning

confidence: 94%

“…Although the term Moran particle system is used in [18] and a few other works, most studies concerned with quasi-stationary distributions refer to Fleming-Viot particle system, see e.g. [2,9,11,13,21,23,27,30,31] (for continuous-time processes, which thus corresponds to the process introduced in 2.4 i.e. the limit γ → 0).…”

Section: Related Workmentioning

confidence: 99%

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case

Monmarché¹,

Journel²

2022

ESAIM: PS

View full text Add to dashboard Cite

We establish the convergences (with respect to the simulation time $t$; the number of particles $N$; the timestep $\gamma$) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the $d$-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter ($t\rightarrow \infty$, $N\rightarrow \infty$ or $\gamma\rightarrow 0$) are independent from the two others. p, li { white-space: pre-wrap; }

show abstract

“…However, to guide the reader, we end this section with some references to the literature on mean-field-type particle methodologies currently used in this context. Mean field and genetic-type particle methodologies are discussed in [52,63,64,65,66,68,69], as well as [7,27,28,126,163,181], and [3,4,5,53,54,45,97,176].…”

Section: Literature Reviewmentioning

confidence: 99%

On the Stability of Positive Semigroups

Moral¹,

Horton²,

Jasra³

2021

Preprint

View full text Add to dashboard Cite

The stability and contraction properties of positive integral semigroups on Polish spaces are investigated. We provide a novel analysis based on an extension of V -norm, Dobrushin-type, contraction techniques on functionally weighted Banach spaces for Markov operators. These are applied to a general class of positive and possibly time-inhomogeneous bounded integral semigroups and their normalised versions. Under mild regularity conditions, the Lipschitz-type contraction analysis presented in this article simplifies and extends several exponential estimates developed in the literature. The spectral-type theorems that we develop can also be seen as an extension of Perron-Frobenius and Krein-Rutman theorems for positive operators to time-varying positive semigroups. Incidentally, in the context of time-homogeneous models, the regularity conditions discussed in the present article appears to be a necessary and sufficient condition for the existence of leading eigenvalues. We review and illustrate in detail the impact of these results in the context of positive semigroups arising in transport theory, physics, mathematical biology and advanced signal processing.

show abstract

Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution

Cited by 7 publications

References 25 publications

The asymptotic tails of limit distributions of continuous time Markov chains

The asymptotic tails of limit distributions of continuous time Markov chains

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case

On the Stability of Positive Semigroups

Contact Info

Product

Resources

About