Parametrix techniques and martingale problems for some degenerate Kolmogorov equations

Stephane Menozzi

doi:10.1214/ecp.v16-1619

Abstract

We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [BP09] to prove uniqueness of the martingale problem in the framework of non- degenerate elliptic operators and the Mc Kean and Singer [MS67] parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi [DM10]).

Highlights

1.1 Martingale problem and parametrix techniquesThe martingale approach turns out to be useful when trying to get uniqueness results for the stochastic process corresponding to an operator
We prove the uniqueness of the martingale problem associated to some degenerate operators
The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [2] to prove uniqueness of the martingale problem in the framework of non-degenerate elliptic operators and the Mc Kean and Singer [13] parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider

Summary

Martingale problem and parametrix techniques

The martingale approach turns out to be useful when trying to get uniqueness results for the stochastic process corresponding to an operator. Considering a sequence of equations with mollified coefficients, we derive from convergence in law, the RadonNikodym theorem and (1.4) that the unique weak solution of d X t = σ(X t )dWt associated to L admits a density that satisfies the previous Gaussian bound. We used this technique to give a local limit theorem for the Markov chain approximation of a Langevin process [12] or two-sided bounds of some more general degenerate hypoelliptic operators [3] In both works, we have an unbounded drift term. The strategy of the previous articles allows to extend the technique of Bass and Perkins to prove uniqueness of the martingale problem for some degenerate operators with unbounded coefficients

Statement of the Problem and Main Results

Choice of the Gaussian process for the parametrix

Some estimates on degenerate Gaussian processes with linear drift

Linearization of the initial dynamics and associated estimates