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A Version of Hörmander’s Theorem for Markovian Rough Paths

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Abstract

We consider a rough differential equation of the form dYt = ∑ iVi(Yt)dXti + V0(Yt)dt, where Xt is a Markovian rough path. We demonstrate that if the vector fields (Vi)0≤i≤d satisfy the parabolic Hörmander’s 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 a non-degenerate property of the Jacobian process of Xt.

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Acknowledgements

The author is grateful to Fabrice Baudoin for many insightful discussions.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Guang Yang

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  1. Guang Yang
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Correspondence to Guang Yang.

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Partial financial support was received from National Science Foundation grant DMS-1901315.

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Yang, G. A Version of Hörmander’s Theorem for Markovian Rough Paths. Potential Anal 60, 173–195 (2024). https://doi.org/10.1007/s11118-022-10046-5

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