Parametrix and local limit theorem for some degenerate diffusions

Stochastics Seminar
Friday, October 31, 2008 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 255
Valentin Konakov – CEMI RAS, Moscow and UNCC, Charlotte
Heinrich Matzinger

Consider a class of multidimensional degenerate diffusion processes of the following form
X_t = x+\int_0^t (X_s) ds+\int_0^t \sigma(X_s) dW_s,
Y_t = y+\int_0^t F(X_s)ds,
where b,\sigma, F are assumed to be smooth and b,\sigma bounded. Suppose now that \sigma\sigma^* is uniformly elliptic and that \nabla F does not degenerate. These assumptions guarantee that only one Poisson bracket is needed to span the whole space. We obtain a parametrix representation of Mc Kean-Singer type for the density of (X_t,Y_t) from which we derive some explicit Gaussian controls 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 "weak" degeneracy allows to use the local limit Theorem in Gaussian regime but also induces some difficulty to define the suitable approximating process. In particular two time scales appear. Another difficulty w.r.t. the standard literature on the topic, see e.g. Konakov and Mammen (2000), is the unboundedness of F.