Explicit parametrix and local limit theorems for some degenerate diffusion processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 908-923.

Pour une classe de processus de diffusion de rang deux, i.e. lorsque seuls des crochets de Poisson d'ordre un permettent d'engendrer l'espace, nous obtenons une représentation parametrix de type McMean-Singer [J. Differential Geom. 1 (1967) 43-69] de la densité. Nous en dérivons une borne supérieure gaussienne explicite et une borne inférieure partielle qui caractérisent la singularité additionnelle induite par la dégénérescence. Nous donnons ensuite un théorème limite local pour une approximation par chaîne de Markov associée. Le point crucial est que la faible dégénérescence permet d'exploiter les techniques initialement introduites par Konakov et Molchanov [Teor. Veroyatn. Mat. Statist. 31 (1984) 51-64] puis développées dans [Probab. Theory Related Fields 117 (2000) 551-587] et qui reposent sur des approximations gaussiennes.

For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean-Singer [J. Differential Geom. 1 (1967) 43-69] type for the density. We therefrom derive an explicit gaussian upper bound and a partial lower bound that characterize the additional singularity induced by the degeneracy. This particular representation then allows to give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the “weak” degeneracy allows to exploit the techniques first introduced in Konakov and Molchanov [Teor. Veroyatn. Mat. Statist. 31 (1984) 51-64] and then developed in [Probab. Theory Related Fields 117 (2000) 551-587] that rely on gaussian approximations.

DOI : 10.1214/09-AIHP207
Classification : 60J35, 60J60, 35K65
Mots-clés : degenerate diffusion processes, parametrix, Markov chain approximation, local limit theorems 
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Konakov, Valentin; Menozzi, Stéphane; Molchanov, Stanislav. Explicit parametrix and local limit theorems for some degenerate diffusion processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 908-923. doi : 10.1214/09-AIHP207. http://archive.numdam.org/articles/10.1214/09-AIHP207/

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