Explicit parametrix and local limit theorems for some degenerate diffusion processes

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 ofMcKean-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.