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

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 $X_t$.