arXiv subjects

S. Méléard

Publications and source records attributed to S. Méléard.

4 recordsLinked to original sources

arXiv2022

Large population limit of the spectrum of killed birth-and-death processes

We consider a general class of birth-and-death processes with state space $\{0,1,2,3,\ldots\}$ which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at $0$ in the large population limit, that is, we scale the process by a parameter $K$, and take the limit $K\to+\infty$. We assume that the differential equation $\mathrm{d} x/\mathrm{d} t=b(x)-d(x)$ describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at $0$, and an attractive fixed point $x_*>0$. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is $n(b'(x_*)-d'(x_*))$, $n\ge 0$. The other one is the spectrum of a continuous-time binary branching process conditioned on non-extinction, and is given by $n(d'(0)-b'(0))$, $n\ge 1$. A major difficulty is that different scales and function spaces are involved. We work at the level of the eigenfunctions that we split over different regions, and study their asymptotic dependence on $K$ in each region. In particular, we prove that the spectral gap goes to $\min\big\{b'(0)-d'(0),\,d'(x_*)-b'(x_*)\big\}$. This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.

math.PR↗

arXiv2020

Quasi-Stationary Distributions and Resilience: What to get from a sample?

We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by $K$ and in the limit $K\to+\infty$, the realizations of such processes get close, in any fixed finite-time window, to the trajectories of a dynamical system whose vector field is defined by the birth and death rates. Assuming that this dynamical has a unique attracting fixed point, we analyzed in a previous work what happens for large but finite $K$, especially the different time scales showing up. In the present work, we are mainly interested in the following question: Observing a realization of the process, can we determine the so-called engineering resilience? To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities. Analogous relations are well known in nonequilibrium statistical mechanics. To exploit these relations, we need to introduce several estimators which we control for times between $\log K$ (time scale to converge to the qsd) and $\exp(K)$ (time scale of mean time to extinction).

math.PR↗

arXiv2018

On time scales and quasi-stationary distributions for multitype birth-and-death processes

We consider a class of birth-and-death processes describing a population made of $d$ sub-populations of different types which interact with one another. The state space is $\mathbb{Z}_+^d$ (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter $K$ which can be thought as the order of magnitude of the total size of the population at time $0$. For any fixed finite time span, it is well-known that such processes, when renormalized by $K$, are close, in the limit $K\to+\infty$, to the solutions of a certain differential equation in $\mathbb{R}_+^d$ whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for $K$ large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time $t$ and the qsd. This bound is exponentially small in $t$, for $t\gg \log K$. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than $\log K$ and much smaller than the mean time to extinction, which is exponentially large as a function of $K$. Let us stress that we are interested in what happens for finite $K$. We obtain results much beyond what large deviation techniques could provide.

math.PR↗

arXiv2014

Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

We study a general class of birth-and-death processes with state space $\mathbb{N}$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a `carrying capacity' $K$. When $K$ is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of $K$, for large $K$. We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when $K$ is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process.

math.PR↗