Quasi-stationary Distribution Research Articles

Recent advances in the long-time analysis of killed degenerate processes and their particle approximation

We review some recent results of quantitative long-time convergence for the law of a killed Markov process conditioned to survival toward a quasi-stationary distribution, and on the analogous question for the particle systems used in practice to sample these distributions. With respect to the existing literature, one of the novelties of these works is the degeneracy of the underlying process with respect to classical elliptic diffusion, namely it can be a non-elliptic hypoelliptic diffusion, a piecewise deterministic Markov process or an Euler numerical scheme.

Existence of quasi-stationary distributions for spectrally positive Lévy processes on the half-line

Existence of quasi-stationary distributions for spectrally positive Lévy processes on the half-line

General criteria for the study of quasi-stationarity

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

Exponential quasi-ergodicity for processes with discontinuous trajectories

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’s inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property that 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 ℝd that do not satisfy the classical Harnack’s inequality, 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.

Transcritical Bifurcation for the Conditional Distribution of a Diffusion Process

In this article, we describe a simple class of models of absorbed diffusion processes with parameter, whose conditional law exhibits a transcritical bifurcation. Our proofs are based on the description of the set of quasi-stationary distributions for general two-clusters reducible processes.

Asymptotic behavior and quasi-limiting distributions on time-fractional birth and death processes

AbstractIn this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23].

On the Hill relation and the mean reaction time for metastable processes

We illustrate how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the biasing error introduced by many numerical procedures that have been proposed in the literature, in particular in molecular dynamics, to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the biasing error analysis as well as the applicability of our study to elliptic diffusions.

Cross-graph reference structure based pruning and edge context information for graph matching

Graph matching (GM) plays a vital role in solving various computer vision tasks by establishing a node-to-node correspondence between graph-structured data. In this study, we propose a novel method based on affinity matrix for solving GM tasks. Specifically, a pruning strategy is proposed to remove inconsistent edges from an association graph that formulates the GM problem into a node ranking and selection problem. Moreover, an edge context information with respect to a cross-graph reference structure is proposed and combined with the local distance information to compute the affinities among the graphs’ nodes. Then, the re-weighted random walk technique is utilized to simulate random walks on the association graph and compute a quasi-stationary distribution in an iterative manner. Finally, the Hungarian algorithm is used to discretize the distribution and obtain an approximate matching between two graphs. The experimental results indicate that our proposed GM method is able to produce promising results in the presence of outliers and deformation noise as compared with other methods.

Noise-induced sustainability of cooperation in Prisoner's Dilemma game

Based on the concept of stochastic Prisoner’s Dilemma (SPD) game in experimental economics proposed by Bereby-Meyer and Roth (Am Econ Rev 96 (2006) 1029–1042) (see also H. Kunreuther et al., Judgm. Decis. Mak. 4 (5) (2009) 363–384), the dynamical properties of the SPD game is investigated in this paper. For the non-repeated SPD game, we can see that the increase of noise intensity will not only lead to loss of the stochastic stability of the boundary state corresponding to defection but also the non-equilibrium phase transition of the quasi-stationary distribution of the system, and that the fixation probability of cooperation will increase with the increase of noise intensity. However, for the stochastic TFT-AllD game, the increase of noise intensity will lead to the loss of the stochastic stability of the two boundaries corresponding to cooperation and defection respectively, but will not lead to the non-equilibrium phase transition of the quasi-stationary distribution of the system. Moreover, as a special case of the stochastic TFT-AllD game (i.e., (m−1)(b−c)=2c), we show that the fixation probability of TFT will increase (decrease) with the increase of noise intensity if the initial frequency of TFT is less (or larger) than 1/2.

Domain of attraction of quasi-stationary distribution for absorbing Markov processes

In this paper, for the general λ0-positive absorbing Markov processes, we obtain a concise result on the domain of attraction of the quasi-stationary distribution. Our conclusion includes some existing results. Finally, we apply our result to multi-dimensional Ornstein–Uhlenbeck process and give a subset of the domain of attraction of its minimal quasi-stationary distribution.

Uniform in time propagation of chaos for a Moran model

This article studies the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model. Our results include a uniform in time bound for the propagation of chaos in Lp of order N, and the proof of the asymptotic normality with zero mean and explicit variance, when the number of individuals tend towards infinity, for the approximation error between the empirical distribution and its limit. Additionally, we explore the interpretation of this Moran model as a particle process whose empirical probability measure approximates a quasi-stationary distribution, in the same spirit as the Fleming–Viot particle systems.

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

AbstractThis paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

Exponential convergence to a quasi-stationary distribution for birth–death processes with an entrance boundary at infinity

AbstractWe study the quasi-stationary behavior of the birth–death process with an entrance boundary at infinity. We give by the h-transform an alternative and simpler proof for the exponential convergence of conditioned distributions to a unique quasi-stationary distribution in the total variation norm. In addition, we also show that starting from any initial distribution the conditional probability converges to the unique quasi-stationary distribution exponentially fast in the $\psi$ -norm.

A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions

AbstractWe study convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions. We give a method for reducing the convergence to the tail behavior of the lifetime via a property we call the first hitting uniqueness. We apply the results to Kummer diffusions with negative drift and give a class of initial distributions converging to each non-minimal quasi-stationary distribution.

Mathematical foundations for the Parallel Replica algorithm applied to the underdamped Langevin dynamics

Molecular dynamics (MD) methods are used to sample the time evolution of complex molecular systems, namely the transition events between configuration states. When these states are metastable, these transitions correspond to rare events and occur over very longtime scale, thus rendering the use of MD methods inefficient. The Parallel Replica algorithm, was designed to efficiently sample these rare events relying on a parallelization in time computation of the trajectory. Its mathematical formalization was carried out in the literature using the notion of quasi-stationary distribution (QSD), which can be seen as the longtime distribution of the dynamics trapped inside a state. Its existence was only proven recently for the underdamped Langevin dynamics (ULD) involved in the sampling of thermostated molecular systems. In this research letter we shall state the existence of a QSD for the ULD, for which we can provide a spectral interpretation. The existence of this QSD is then used to justify the application of Parallel Replica algorithm to the ULD. In addition, we shall extend known results regarding the behaviour of the ULD trajectories when the friction coefficient goes to infinity (overdamped limit). This allows us to provide an explicit expression of the overdamped limit of the previous QSD of the ULD. The findings of this study remain valid for ULD with non-conservative forces. They shall help for a better understanding of the mathematical framework underlying the Parallel Replica algorithm or related algorithms involving the QSD to the ULD. It also extends our knowledge on the overdamped limiting behaviour of the ULD.Graphical abstract

Quasi-stationary distribution for Hamiltonian dynamics with singular potentials

In this work, we prove the existence and the uniqueness of a quasi-stationary distribution for hypoelliptic Hamiltonian dynamics for a system of N particles in $${\mathbb {R}}^d$$ interacting with Lennard-Jones type potentials or with repulsive Coulomb potentials.

The stationary and quasi-stationary properties of neutral multi-type branching process diffusions

The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Pedagogical derivations are given for known results that the limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with discrete multi-type branching processes. For the sub-critical process the previously unsolved quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size.

A non‐conservative Harris ergodic theorem

We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of the spectral gap, complementing Krein–Rutman theorems and generalizing probabilistic approaches. The proof is based on a non-homogenous h $h$ -transform of the semigroup and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris's theorem for conservative semigroups and recent techniques developed for the study of absorbed Markov processes. We apply these results to population dynamics. We obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs.

Broadband and photovoltaic THz/IR response in the GaAs-based ratchet photodetector.

High-performance broadband infrared (IR)/terahertz (THz) detection is crucial in many optoelectronic applications. However, the spectral response range of semiconductor-based photodetectors is limited by the bandgaps. This paper proposes a ratchet structure based on the GaAs/AlxGa1−xAs heterojunction, where the quasi-stationary hot hole distribution and intravalence band absorption from light or heavy hole states to the split-off band overcome the bandgap limit, ensuring an ultrabroadband photoresponse from near-IR to THz region (4 to 300 THz). The peak responsivity of the proposed structure can reach 7.3 A/W, which is five orders of magnitude higher than that of the existing broadband photon-type detector. Because of the ratchet effect, the proposed photodetector has a bias-tunable photoresponse characteristic and can operate in the photovoltaic mode with a broad photocurrent spectrum (18 to 300 THz). This work not only demonstrates a broadband photon-type THz/IR photodetector but also provides a method to study the light-responsive ratchet.

Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). We study the flow of ancestral lineages as time goes to the past and show that, properly renormalized, it converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. The inverse subordinator is shown to be partitioning the current population into ancestral families with distinct common ancestors. When Grey’s condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families have i.i.d. sizes distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey’s condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points S of the limiting inverse subordinator. The proof is based on a general result for stochastically monotone processes of independent interest, which relates θ-invariant measures and θ-invariant functions for a process and its Siegmund dual.