We define a notion of coupling time with ambiguities for interacting particle systems, and show how this can be used to prove ergodicity and to bound the convergence time to equilibrium and the decay of correlations at equilibrium. A motivation is to provide simple conditions which ensure that perturbed particle systems share some properties of the underlying unperturbed system. We apply these results to context-dependent substitution models recently introduced by molecular biologists as descriptions of DNA evolution processes. These models take into account the influence of the neighboring bases on the substitution probabilities at a site of the DNA sequence, as opposed to most usual substitution models which assume that sites evolve independently of each other.
This paper is devoted to interacting particle systems on the integer line Z with finite state space S, whose dynamics is characterized by a finite list R of stochastic transition rules. We now give an informal description of the dynamics that we consider for these systems, and we postpone a proper mathematical definition to section 2.
1.1. Construction of interacting particle systems dynamics. We begin with some vocabulary. A state s is an element of S, a site x is an element of Z, a configuration ξ := (ξ(x)) x∈Z is an element of S Z . A rule R := (c, r) is based on a context c and characterized by a rate r. A context is a triple c := (A, ℓ, s), where A is a finite subset of Z, ℓ is a subset of S A , s is a state, and r is a rate, that is, a non-negative real number.
We say that a configuration ξ and a context c = (A, ℓ, s), or any rule R = (c, r) based on c, are compatible at site x if A is empty, or if A is not empty and ξ(x + A) belongs to ℓ, where ξ(x + A) is the element of S A defined as ξ(x + A) := (ξ(x + y)) y∈A .
The interacting particle system is a Markov process (X t ) t on S Z whose dynamics is characterized by a given finite list R := (R i ) i∈I of stochastic transition rules, as follows: for any time t, if a rule R i = (c i , r i ) in R with c i = (A i , ℓ i , s i ) is compatible with X t at site x, then X t+dt (x) = s i with probability r i dt + o(dt), independently of every other rule in R, compatible with X t at site x or elsewhere.
A classical way to give a more explicit construction of such particle systems uses the so-called graphical representation (see for instance [11] page 142 for a discussion in the context of voter models). This amounts to a stochastic flow based on Poisson processes: given a time t and an initial condition ξ in S Z imposed at time t, the Poisson processes determine the state of the particle system at every time greater than t. Once again informally, to every site x and rule R i = (c i , r i ) in R corresponds a homogenous Poisson process Ψ(x, i) on the real line R with rate r i , and the points of Ψ(x, i) are the random times at which the rule R i is applied to the state at site x. Specifically, for every rule R i = (c i , r i ) in R with context c i = (A i , ℓ i , s i ), if t belongs to Ψ(x, i) and if R i and X tare compatible at site x, then X t (x) = s; otherwise, X t (x) = X t-(x). See section 2 for a proper definition.
1.2. Coupling times. Within this framework, various notions of coupling times can be defined. In this paper, an ordinary coupling time is an almost surely finite random variable T with negative values, measurable with respect to the family (Ψ(x, i)) (x,i)∈Z×I of Poisson processes, and such that, for every time u < T , if the dynamics starts at time u, the state of site x = 0 at time t = 0 -is the same for every initial condition at time u. This definition corresponds to a coupling from the past, as opposed to the usual notion of forward coupling.
As soon as such coupling times exist, the particle system is ergodic. Furthermore, estimates on the tail of T yield estimates on the rate of convergence to equilibrium, and additional assumptions on the coupling time yield estimates on the decay of correlations. Consider now the set of points
where the union runs over every x in Z and i in I. A point in T corresponds to a transition that may or may not be performed between the times t = T and t = 0 -, depending on the initial condition at time v < T . When, for a given (u, x) in T , there indeed exists v < T and two distinct initial conditions at time v such that, for one of these initial conditions, the transition proposed by (u, x) is performed, while it is not performed when the other initial condition is used, we say that an ambiguity arises at (u, x). By the definition of an ordinary coupling time, one sees that, for each time in T , either there is no ambiguity associated with it, or there is an ambiguity that has no influence on the state of site x = 0 at time t = 0 -. We can now define, once again informally, the notion of coupling time with ambiguities. This is a pair (H, T ), where T is a random variable with negative values, measurable with respect to the family (Ψ(x, i)) (x,i)∈Z×I of Poisson processes and H is a finite random subset of the set T defined above, enjoying the stopping property, and such that the following property holds: for any two initial conditions at time u < T such that the ambiguities associated with the elements of H are resolved in the same way (that is, a transition corresponding to an element of H is either performed for both initial conditions, or not performed for both initial conditions), the state of site x = 0 at time t = 0 -is the same for both initial conditions. One sees that, if (H, T ) is a coupling time with ambiguities, T may or may not be an ordinary coupling time. However, the only ambiguities that may prevent T from being an ordinary coupling time are those associated to the points in H. As a con
This content is AI-processed based on open access ArXiv data.
