Parametrix techniques and martingale problem for some degenerate Kolmogorov's equations

TL;DR

This paper establishes the uniqueness of the martingale problem for certain degenerate Kolmogorov operators by combining parametrix techniques with a novel approach inspired by Bass and Perkins, extending previous density expansion methods.

Contribution

It introduces a new method that merges parametrix techniques with a probabilistic approach to prove uniqueness for degenerate operators, building on prior work in non-degenerate and degenerate cases.

Findings

Proves uniqueness of the martingale problem for specific degenerate operators.

Extends parametrix density expansion techniques to degenerate settings.

Provides a new framework combining probabilistic and analytical methods.

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 (From Probability to Geometry, vol. in honor of J.M Bismut (2009)) to prove uniqueness of the martingale problem in the framework of non degenerated elliptic operators and the Mc Kean and Singer (Journal of Diff. Geometry, 1967) parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi, Journal of Funct. Analysis, 2010).