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Register a new userConvergence Time to Equilibrium of the Metropolis dynamics for the GREM
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We study the convergence time to equilibrium of the Metropolis dynamics for the Generalized Random Energy Model with an arbitrary number of hierarchical levels, a finite and reversible continuous-time Markov process, in terms of the spectral gap of its transition probability matrix. This is done by deducing bounds to the inverse of the gap using a Poincare inequality and a path technique. We also apply convex analysis tools to give the bounds in the most general case of the model.
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The main objective of the paper is to study the long-time behavior of general discrete dynamics driven by an ergodic stationary Gaussian noise. In our main result, we prove existence and uniqueness of the invariant distribution and exhibit some upper-bounds on the rate of convergence to equilibrium in terms of the asymptotic behavior of the covariance function of the Gaussian noise (or equivalently to its moving average representation). Then, we apply our general results to fractional dynamics (including the Euler Scheme associated to fractional driven Stochastic Differential Equations). Whenthe Hurst parameter H belongs to (0, 1/2) we retrieve, with a slightly more explicit approach due to the discrete-time setting, the rate exhibited by Hairer in a continuous time setting. In this fractional setting, we also emphasize the significant dependence of the rate of convergence to equilibriumon the local behaviour of the covariance function of the Gaussian noise.
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
Fluid models have become an important tool for the study of many-server queues with general service and patience time distributions. The equilibrium state of a fluid model has been revealed by Whitt (2006) and shown to yield reasonable approximations to the steady state of the original stochastic systems. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under a mild condition. Our method builds on the framework of measure-valued processes developed in Zhang (2013), which keeps track of the remaining patience and service times.
The supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of the freezing rate for a wide range of initial temperature distributions in the liquid. Such a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we prove that the natural Euler time-stepping scheme applied to a probabilistic formulation of the supercooled Stefan problem converges to the liquid-solid boundary of its physical solution globally in time, in the Skorokhod M1 topology. In the course of the proof, we give an explicit bound on the rate of local convergence for the time-stepping scheme. We also run numerical tests to compare our theoretical results to the practically observed convergence behavior.
We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the {em special endogenous solution} to a stochastic fixed-point equation of the form: $$Rstackrel{mathcal D}{=} Phi( Q, N, { C_i }, {R_i}),$$ where $(Q, N, {C_i})$ is a real-valued random vector with $N in mathbb{N}$, and ${R_i}_{i in mathbb{N}}$ is a sequence of i.i.d. copies of $R$, independent of $(Q, N, {C_i})$; the symbol $stackrel{mathcal{D}}{=}$ denotes equality in distribution. Specifically, we show its convergence in the Wasserstein metric of order $p$ ($p geq 1$) and prove the consistency of estimators based on the sample pool produced by the algorithm.
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