A Version of Hormander's Theorem for Markovian Rough Paths

We consider a rough differential equation of the form dY(t) = n-ary sumation V-i(i)(Y-t)dXti + V-0(Y-t)dt, where X-t is a Markovian rough path. We demonstrate that if the vector fields (V-i)(0 <= i <= d) satisfy the parabolic Hormander's 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 a non-degenerate property of the Jacobian process of X-t.