Dynamics of a Fleming–Viot type particle system on the cycle graph

Corujo, Josué

doi:10.1016/j.spa.2021.02.001

Cited by 2 publications

(3 citation statements)

References 22 publications

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“…See e.g. [CT16b] and [Cor21a]. We will prove that this result also holds when V is symmetric, (see Corollary 3.6 below).…”

Section: Introduction and Main Resultsmentioning

confidence: 72%

Uniform in time propagation of chaos for a Moran model

Cloez,

Corujo

2021

Preprint

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The goal of this article is to study the limit of the empirical distribution induced by a multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. Throughout the paper, the mutation rate driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided in two parts. The first one deals with processes with general selection rates. For this case we are able to prove the propagation of chaos in L p over the compacts with speed of convergence of order 1/ √ N . Further on, we will consider a specific type of selection that we will call additive selection. Essentially, we assume that the selection rate can be decomposed in the sum of three elements: a term depending on the allelic type of the parent (which can be understood as selection at death), another term depending on the allelic type of the descendant (which can be understood as selection at birth) and a third term which is symmetric. Under this setting, our results include a uniform in time bound for the propagation on chaos in L p of order 1/ √ N , and an asymptotic expression for the quadratic error between the empirical distribution and its limits when both, the time and the number of individuals, tend towards infinity. Additionally, we explore the interpretation of the Moran model with additive selection as a particle process whose empirical distribution approximate a quasi-stationary distribution, in the same spirit of the Fleming -Viot particle systems. We then address to the problem of minimising the asymptotic quadratic error, when the time and the number of particles go to infinity. Contents 1. Introduction and main results 1 1.1. Main results 2 Structure of the paper 9 2. Links to the theory of QSD 9 2.1. Examples 10 2.2. Discussion 12 3. Proof of the main results 13 3.1. The associated martingale problem 13 3.2.

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“…See e.g. [CT16b] and [Cor21a]. We will prove that this result also holds when V is symmetric, (see Corollary 3.6 below).…”

Section: Introduction and Main Resultsmentioning

confidence: 72%

Uniform in time propagation of chaos for a Moran model

Cloez,

Corujo

2021

Preprint

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“…The following example uses Corollary 1.6 to provide the rates for the exponential convergence to stationarity of the neutral multi-allelic Moran process considered by Corujo (2021).…”

Section: Applications To the Convergences To Stationaritymentioning

confidence: 99%

“…Q θ is the infinitesimal generator of a simple asymmetric random walk on the Kcycle graph. The neutral multi-allelic Moran type process with mutation rate Q θ was considered in (Corujo, 2021). Since Q θ is circulant, it is possible to explicitly diagonalise it using the Fourier matrix.…”

Section: Applications To the Convergences To Stationaritymentioning

confidence: 99%

On the spectrum and ergodicity of a neutral multi-allelic Moran model

Corujo¹

2023

ALEA

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The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuoustime Markov chain with an unknown stationary distribution. Specifically, we consider N individuals such that each one of them is of one type among K possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.

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Dynamics of a Fleming–Viot type particle system on the cycle graph

Cited by 2 publications

References 22 publications

Uniform in time propagation of chaos for a Moran model

Uniform in time propagation of chaos for a Moran model

On the spectrum and ergodicity of a neutral multi-allelic Moran model

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