A Version of Hörmander's Theorem for Markovian Rough Paths

2020/05/19 by Yang, Guang
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2005.09192

Abstract

We consider a rough differential equation of the form \(dY_t=∑_i V_i(Y_t)d\boldsymbolXⁱ_t+V₀(Y_t)dt \), where \(\boldsymbolX_t \) is a Markovian rough path. We demonstrate that if the vector fields \((V_i)_{0≤ i≤ d} \) satisfy Hörmander'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 \(X_t\).

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