A Version of H\"ormander's Theorem for Markovian Rough Paths
Abstract
We consider a rough differential equation of the form \(dYt=Σi Vi(Yt)dXit+V0(Yt)dt \), where \(Xt \) is a Markovian rough path. We demonstrate that if the vector fields \((Vi)0≤ i≤ d \) satisfy H\"ormander's bracket generating condition, then \(Yt\) admits a smooth density with a Gaussian type upper bound, given that the generator of \(Xt\) satisfy certain non-degenerate conditions. The main new ingredient of this paper is the study of non-degenerate property of the Jacobian process of \(Xt\).
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