Explicit parametrix and local limit theorems for some degenerate diffusion processes

Valentin Konakov; Stéphane Menozzi; Stanislav Molchanov

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 908-923
  • ISSN: 0246-0203

Abstract

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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 Fields117 (2000) 551–587] that rely on gaussian approximations.

How to cite

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Konakov, Valentin, Menozzi, Stéphane, and Molchanov, Stanislav. "Explicit parametrix and local limit theorems for some degenerate diffusion processes." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 908-923. <http://eudml.org/doc/241286>.

@article{Konakov2010,
abstract = {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 Fields117 (2000) 551–587] that rely on gaussian approximations.},
author = {Konakov, Valentin, Menozzi, Stéphane, Molchanov, Stanislav},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {degenerate diffusion processes; parametrix; Markov chain approximation; local limit theorems},
language = {eng},
number = {4},
pages = {908-923},
publisher = {Gauthier-Villars},
title = {Explicit parametrix and local limit theorems for some degenerate diffusion processes},
url = {http://eudml.org/doc/241286},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Konakov, Valentin
AU - Menozzi, Stéphane
AU - Molchanov, Stanislav
TI - Explicit parametrix and local limit theorems for some degenerate diffusion processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 908
EP - 923
AB - 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 Fields117 (2000) 551–587] that rely on gaussian approximations.
LA - eng
KW - degenerate diffusion processes; parametrix; Markov chain approximation; local limit theorems
UR - http://eudml.org/doc/241286
ER -

References

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