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COUPLING TIMES WITH AMBIGUITIES FOR PARTICLE SYSTEMS AND APPLICATIONS TO CONTEXT DEPENDENT

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Niveau: Supérieur, Doctorat, Bac+8
COUPLING TIMES WITH AMBIGUITIES FOR PARTICLE SYSTEMS AND APPLICATIONS TO CONTEXT-DEPENDENT DNA SUBSTITUTION MODELS JEAN BERARD, DIDIER PIAU Abstract. We define a notion of coupling time with ambiguities for interact- ing 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 un- perturbed 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 neigh- boring 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. Contents 1. Introduction and motivations 1 2. Formal setting 4 3. Statement of the main results 10 4. Proof of the main results 11 5. Applications to nucleotide substitution models 20 References 32 1. Introduction and motivations 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.

  • sz

  • based

  • sample space

  • coupling time

  • particle system

  • markov process

  • markov chains


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COUPLING TIMES WITH AMBIGUITIES FOR PARTICLE SYSTEMS AND APPLICATIONS TO CONTEXT-DEPENDENT DNA SUBSTITUTION MODELS
´ JEAN BERARD, DIDIER PIAU
Abstract.We define a notion of coupling time with ambiguities for interact-ing 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 un-perturbed 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 neigh-boring 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.
Contents
1. Introduction and motivations
2. Formal setting
3. Statement of the main results
4. Proof of the main results
5. Applications to nucleotide substitution models
References
1.Introduction and motivations
1
4
10
11
20
32
This paper is devoted to interacting particle systems on the integer lineZwith finite state spaceS, whose dynamics is characterized by a finite listRof 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.
Date: November 2, 2007. 2000Mathematics Subject Classification.60J25, 60K35, 92D20. Key words and phrases.Interacting particle systems, Coupling, Perturbations, Stochastic models of nucleotide substitutions. 1
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´ JEAN BERARD, DIDIER PIAU
1.1.Construction of interacting particle systems dynamics.We begin with some vocabulary. Astatesis an element ofS, asitexis an element ofZ, a configurationξ:= (ξ(x))xZis an element ofSZ. AruleR:= (c, r) is based on a contextcand characterized by a rater. Acontextis a triplec:= (A, `, s), where Ais a finite subset ofZ,`is a subset ofSA,sis a state, andris arate, that is, a non-negative real number. We say that a configurationξand a contextc= (A, `, s), or any ruleR= (c, r) based onc, arecompatible at sitexifAis empty, or ifAis not empty andξ(x+A) belongs to`, whereξ(x+A) is the element ofSAdefined as ξ(x+A) := (ξ(x+y))yA. The interacting particle system is a Markov process (Xt)tonSZwhose dynamics is characterized by a given finite listR:= (Ri)iIof stochastic transition rules, as follows: for any timet, if a ruleRi= (ci, ri) inRwithci= (Ai, `i, si) is compatible withXtat sitex, thenXt+dt(x) =siwith probabilityridt+o(dt), independently of every other rule inR, compatible withXtat sitexor 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 timetand an initial conditionξinSZimposed at timet, the Poisson processes determine the state of the particle system at every time greater thant again informally, to every site. Oncexand ruleRi= (ci, ri) inRcorresponds a homogenous Poisson process Ψ(x, i) on the real lineRwith rateri, and the points of Ψ(x, i) are the random times at which the ruleRiis applied to the state at site x for every rule. Specifically,Ri= (ci, ri) inRwith contextci= (Ai, `i, si), if tbelongs to Ψ(x, i) and ifRiandXtare compatible at sitex, thenXt(x) =s; otherwise,Xt(x) =Xt(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, anordinary coupling timeis an almost surely finite random variableTnegative values, measurable with respect to the familywith (Ψ(x, i))(x,i)Z×Iof Poisson processes, and such that, for every timeu < T, if the dynamics starts at timeu, the state of sitex= 0 at timet= 0is the same for every initial condition at timeu definition corresponds to a coupling from. This 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 ofTyield 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 T:=[(Ψ(x, i)[T ,0[)× {x}, (x,i) where the union runs over everyxinZandiinI. A point inTcorresponds to a transition that may or may not be performed between the timest=Tandt= 0, depending on the initial condition at timev < T. When, for a given (u, x) inT, there indeed existsv < Tand two distinct initial conditions at timevsuch 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
COUPLING TIMES WITH AMBIGUITIES FOR PARTICLE SYSTEMS
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ambiguity arises at (u, x). By the definition of an ordinary coupling time, one sees that, for each time inT, either there is no ambiguity associated with it, or there is an ambiguity that has no influence on the state of sitex= 0 at timet= 0. We can now define, once again informally, the notion of coupling time with ambi-guities. This is a pair (H, T), whereTis a random variable with negative values, measurable with respect to the family (Ψ(x, i))(x,i)Z×Iof Poisson processes andH is a finite random subset of the setTdefined above, enjoying the stopping property, and such that the following property holds: for any two initial conditions at time u < Tthe ambiguities associated with the elements ofsuch that Hare resolved in the same way (that is, a transition corresponding to an element ofHis either performed for both initial conditions, or not performed for both initial conditions), the state of sitex= 0 at timet= 0is the same for both initial conditions. One sees that, if (H, T) is a coupling time with ambiguities,Tmay or may not be an ordinary coupling time. However, the only ambiguities that may preventT from being an ordinary coupling time are those associated to the points inH. As a consequence, in the degenerate case whenHis empty,Tis indeed an ordinary coupling time. Informally, our main result is that, if the random setHcontains few enough points on average (we callsubcriticalany coupling time with ambiguities enjoying this property), it is possible to build an ordinary coupling time from (H, T), thus prov-ing ergodicity of the particle system. Moreover, more specific estimates and as-sumptions about the setHprovide estimates on this ordinary coupling time, that are suitable to study the rate of convergence to equilibrium of the particle system and the decay of its correlations. The construction of an ordinary coupling time from a subcritical coupling time with ambiguities is described in section 4. The principle of this construction is to apply iteratively coupling times with ambiguities, looking further and further into the past, until every ambiguity is eventually resolved.
1.3.Perturbed particle systems.We now describe how these results allow to study some perturbed particle systems. We assume that the list of transition rules is of the formR= (Ri)iIoIp, whereIoandIpare disjoint sets, the familyRo:= (Ri)iIobeing the family of so-called non-perturbative rules, whileRp:= (Ri)iIp is the family of so-called perturbative rules. We call the interacting particle system based on the whole family of rulesRthe perturbed system and the system based on the family of non-perturbative rulesRo the unperturbed system. A general problem about perturbations of particle systems is to relate the properties of the perturbed system such as ergodicity, speed of convergence to equilibrium or decay of correlations at equilibrium, to those of the unperturbed system, when the transition rates attached to the perturbative rules are small enough. In this context, we wish to mention two results, one on the negative side and one on the positive side:
(1) Small perturbations of ergodic particle systems may not be ergodic. For a well-known example, consider the two-dimensional Ising model. Its dy-namics is ergodic at the critical inverse temperatureβcand not ergodic at