Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Explicit parametrix and local limit theorems for some degenerate diffusion processes

Valentin Konakov, Stéphane Menozzi, and Stanislav Molchanov

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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.


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.

Article information

Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 4 (2010), 908-923.

First available in Project Euclid: 4 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]
Secondary: 35K65: Degenerate parabolic equations

Degenerate diffusion processes Parametrix Markov chain approximation Local limit theorems


Konakov, Valentin; Menozzi, Stéphane; Molchanov, Stanislav. Explicit parametrix and local limit theorems for some degenerate diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 908--923. doi:10.1214/09-AIHP207.

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