A Version of H\"ormander's Theorem for Markovian Rough Paths

TL;DR

This paper extends Hörmander's theorem to Markovian rough paths, showing that solutions to certain rough differential equations have smooth densities under bracket generating conditions and non-degeneracy assumptions.

Contribution

It establishes a version of Hörmander's theorem for Markovian rough paths, focusing on the non-degenerate Jacobian process to prove smooth density results.

Findings

Solutions have smooth densities with Gaussian bounds

Hörmander's bracket condition ensures regularity

Non-degeneracy of the Jacobian is crucial

Abstract

We consider a rough differential equation of the form \(dY_t=\sum_i V_i(Y_t)d\boldsymbol{X}^i_t+V_0(Y_t)dt \), where \(\boldsymbol{X}_t \) is a Markovian rough path. We demonstrate that if the vector fields \((V_i)_{0\leq i\leq d} \) satisfy H\"ormander's bracket generating condition, then \(Y_t\) admits a smooth density with a Gaussian type upper bound, given that the generator of \(X_t\) satisfy certain non-degenerate conditions. The main new ingredient of this paper is the study of non-degenerate property of the Jacobian process of \(X_t\).