A Version of Hörmander's Theorem for Markovian Rough Paths

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).