Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

Hingant, Erwan; Yvinec, Romain

doi:10.1214/21-ecp411

Cited by 2 publications

(1 citation statement)

References 11 publications

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“…A useful overview which treats discrete problems such as the ones treated here is [5]. A much more probabilistic development of the celebrated Becker-Doring version of the problem is in [36]. This develops rates of convergence using coupling.…”

Section: Random Transpositions and Coagulation-fragmentation Processesmentioning

confidence: 99%

Double coset Markov chains

Diaconis

Ram

Simper

2023

Forum of Mathematics, Sigma

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Let G be a finite group. Let $H, K$ be subgroups of G and $H \backslash G / K$ the double coset space. If Q is a probability on G which is constant on conjugacy classes ( $Q(s^{-1} t s) = Q(t)$ ), then the random walk driven by Q on G projects to a Markov chain on $H \backslash G /K$ . This allows analysis of the lumped chain using the representation theory of G. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains with G a compact group are discussed.

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Section: Random Transpositions and Coagulation-fragmentation Processesmentioning

confidence: 99%

Double coset Markov chains

Diaconis

Ram

Simper

2023

Forum of Mathematics, Sigma

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Exponential quasi-ergodicity for processes with discontinuous trajectories

Velleret

2023

ESAIM: PS

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This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on [[EQUATION]] that do not satisfy the classical Harnack inequalities, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.

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Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

Cited by 2 publications

References 11 publications

Double coset Markov chains

Double coset Markov chains

Exponential quasi-ergodicity for processes with discontinuous trajectories

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