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