Sobolev inequality and the invariance principle for diffusions in periodic potential

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Date/heure
Date(s) - 18/04/2014
11 h 00 min - 12 h 00 min

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We prove here, using stochastic analysis methods ; the invariance principle for a Rd- diffusions d ≥ 2 ; involving in periodic potential beyond uniform boundedness assumptions and beyond regularity assumptions on potential. The potential is not assumed to have any regularity. So the stochastic calculus theory for processes associated to Dirichlet forms used to justify the existence of this process starting for almost all x ∈ Rd. We show by using harmonic analysis ? one Sobolev inequality with different weight to bound the probability of transition associated to the time changed diffusion for all times and deduce the existence of one bounded density. This property allows us to prove easily the tightness of the sequence of processes in the uniform topology. The proof of the con- vergence in finite dimensional distribution is very standard: construction and convergence of the so-called corrector ? and central limit theorem for martingale with continuous time (Helland 1982). The approach used here is the same as in [2] (Mathieu 2008): the notion of time changed process by an additive functional.

http://www.i2m.univ-amu.fr/spip.php?page=pageperso&nom=Ba&prenom=Moustapha“>http://www.i2m.univ-amu.fr/spip.php?page=pageperso&nom=Ba&prenom=Moustapha

Olivier CHABROL
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