Extremal Invariant Measures Research Articles

Phase Transition for Infinite Systems of Spiking Neurons

We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate $1 $ whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all {\it presynaptic neurons} since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to $0$ and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate $\gamma > 0 .$ For this process we prove the existence of a value $\gamma_c$ such that the system has one or two extremal invariant measures according to whether $\gamma > \gamma_c $ or not.

Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input

Nous considerons un modele stochastique decrivant un neurone stimule par un signal qu’il recoit a travers son systeme dendritique. Ce signal aleatoire, de type Ornstein–Uhlenbeck, resulte de la perturbation d’un signal periodique deterministe. Pour decrire l’evolution du potentiel de membrane du neurone, nous faisons appel au modele classique de Hodgkin et Huxley. Le systeme stochastique obtenu est fortement degenere. Nous demontrons que la condition de Hormander faible est satisfaite en tout point de l’espace d’etats. L’existence d’une fonction de Lyapunov permet de conclure qu’il existe au plus un nombre fini de mesures ergodiques pour ce modele. Par consequent, la complexite du systeme est reduite de maniere significative par rapport a la situation deterministe.

Voter model in a random environment in ℤ d

We consider the voter model with flip rates determined by {µ e , e ∈ E d }, where E d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice ℤ d . Suppose that {µ e , e ∈ E d } are independent and identically distributed (i.i.d.) random variables satisfying µ e ⩾ 1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: δ 0 and δ 1.

Some ergodic theorems for a parabolic Anderson model

In this paper, we study some ergodic theorems of a class of linear systems of interacting diffusions, which is a parabolic Anderson model. First, under the assumption that the transition kernel a = (a(i, j))i,j∈S is doubly stochastic, we obtain the long-time convergence to an invariant probability measure νh starting from a bounded a-harmonic function h based on self-duality property, and then we show the convergence to the invariant probability measure νh holds for a broad class of initial distributions. Second, if (a(i, j))i,j∈S is transient and symmetric, and the diffusion parameter c remains below a threshold, we are able to determine the set of extremal invariant probability measures with finite second moment. Finally, in the case that the transition kernel (a(i, j))i,j∈S is doubly stochastic and satisfies Case I (see Case I in [Shiga, T.: An interacting system in population genetics. J. Math. Kyoto Univ., 20, 213–242 (1980)]), we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.

Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of Toom. They are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion. We prove, for a set of initial conditions, exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value. We also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing.

An ergodic theorem of a parabolic Anderson model driven by Lévy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Levy noise. Under the assumption that A = (a(i, j))i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure νh starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Levy noise, which is an extension of the result of Y. Liu and F. X. Yang.

The survival of large dimensional threshold contact processes

We study the threshold θ contact process on ℤd with infection parameter λ. We show that the critical point λc, defined as the threshold for survival starting from every site occupied, vanishes as d→∞. This implies that the threshold θ voter model on ℤd has a nondegenerate extremal invariant measure, when d is large.

An infinite stochastic model of social network formation

We consider an infinite interacting particle system in which individuals choose neighbors according to evolving sets of probabilities. If x chooses y at some time, the effect is to increase the probability that y chooses x at later times. We characterize the extremal invariant measures for this process. In an extremal equilibrium, the set of individuals is partitioned into finite sets called stars, each of which includes a “center” that is always chosen by the other individuals in that set.

An infinite stochastic model of social network formation

We consider an infinite interacting particle system in which individuals choose neighbors according to evolving sets of probabilities. If x chooses y at some time, the effect is to increase the probability that y chooses x at later times. We characterize the extremal invariant measures for this process. In an extremal equilibrium, the set of individuals is partitioned into finite sets called stars, each of which includes a “center” that is always chosen by the other individuals in that set.

Identification of invariant measures of interacting systems

In this paper we provide an approach for identifying certain mixture representations of some invariant measures of interacting stochastic systems. This is related to the problem of ergodicity of certain extremal invariant measures that are translation invariant. Corresponding to these, results concerning the existence of invariant measures and certain weak convergence of the systems are also provided.

Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.

Space–time large deviation lower bounds for spin particle systems

Abstract In this Letter we study large deviation lower bounds for space–time empirical processes of spin particle systems with possibly vanishing spin flip rates. We present an approach for obtaining such lower bounds for systems starting from almost all configurations w.r.t. an translation invariant and extremal invariant measure of the systems, so that in such cases we have full large deviation principle.

On the surviving probability of an annihilating branching process and application to a nonlinear voter model

We study a discrete time interacting particle system which can be considered as an annihilating branching process on Z where at each time every particle either performs a jump as a nearest neighbor random walk, or splits (with probability ε) into two particles which will occupy the nearest neighbor sites. Furthermore, if two particles come to the same site, then they are removed from the system. We show that if the branching probability ε>0 is small enough, and the number of particles at initial time is finite, then the surviving probability, i.e. the probability p( t) that there is at least one particle at time t decays to zero exponentially fast. This result is applied to a nonlinear discrete time voter model (in a random and nonrandom environment) obtained as a small perturbation with parameter ε of the classical voter model. For this class of models, we show that if ε>0 is small enough, then the process converges to a unique invariant probability measure independently on the initial distribution. It is known that the classical one-dimensional voter model (in a random environment as well as without environment) is not ergodic, that is there exist at least two extremal invariant probability measures. Our results prove therefore the phase transition in ε=0.

Large deviations for empirical processes of interacting particle systems

In this paper we study process-level large deviations for spin-flip particle systems and more general Markov processes. We prove that a modification of a well known Donsker-Varadhan entropy function can be used to govern the large deviation lower bounds for any such process starting from a class of initial distributions related to an extremal invariant measure of the process. In some specific cases, we obtain the full large deviation principle.

The Second Lowest Extremal Invariant Measure of the Contact Process II

The Second Lowest Extremal Invariant Measure of the Contact Process II

The second lowest extremal invariant measure of the contact process

We study the ergodic behavior of the contact process on infinite connected graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. In particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence properties of the process, and an associated notion of convergence that we call “partial convergence.” This notion is shown to be better behaved than complete convergence, and to hold in certain cases in which complete convergence fails. Relations between partial and complete convergence are presented, as well as tools to verify when these properties hold. For homogeneous graphs we show that whenever recurrence takes place (i.e., whenever local survival occurs) there are exactly two extremal invariant measures.

A Complete Convergence Theorem for Attractive Reversible Nearest Particle Systems

AbstractIn this paper we prove a complete convergence theorem for attractive, reversible, super-critical nearest particle systems satisfying a natural regularity condition. In particular this implies that under these conditions there exist precisely two extremal invariant measures. The result we prove is relevant to question seven of Liggett (1985), Chapter VII.

Invariant measures of critical spatial branching processes in high dimensions

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the critical Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension dependent. It is known that in low dimensions, $d \leq 2$, the only invariant measure is $\delta_0$ , the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a family ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ of extremal invariant measures; the measures ${\nu_{\theta}$ are translation invariant and indexed by spatial intensity. We prove here, for $d \geq 3$, that all invariant measures are convex combinations of these measures.

Exponential Waiting Time for a Big Gap in a One-Dimensional Zero-Range Process

The first time that the $N$ sites to the right of the origin become empty in a one-dimensional zero-range process is shown to converge exponentially fast, as $N \rightarrow \infty$, to the exponential distribution, when divided by its mean. The initial distribution of the process is assumed to be one of the extremal invariant measures $\nu_\rho, \rho \in (0, 1)$, with density $\rho/(1 - \rho)$. The proof is based on the classical Burke theorem.

On the long term behavior of some finite particle systems

We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ d . Letη t denote this system, and letη denote a finite version ofη t defined on the torus [−N,N] d ∩Z d . Ford≧3 we prove that for stationary, shift ergodic initial measures with density θ, that ifT(N)→∞ andT(N)/(2N+1)d →s∈[0,∞] asN→∞, then {v θ}, θ≧0 is the set of extremal invariant measures for the infinite systemη t andQ s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.