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Explicit parametrix and local limit theorems for some degenerate diffusion processes

Annales de l’Institut Henri Poincaré. 2010. Vol. 46. No. 4. P. 908-923.
Konakov V., Menozzi S., Molchanov S.

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 type for the density. We there from 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 that rely on Gaussian approximations.