Hitting Probabilities for Hypoelliptic Differential Equations Driven by Fractional Brownian Motion
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908697770328064 |
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| author | Geng, Xi Wang, Sheng |
| author_facet | Geng, Xi Wang, Sheng |
| contents | The main goal of this article is to derive a two-sided estimate for hitting probabilities of a hypoelliptic stochastic differential equation (SDE) driven by fractional Brownian motion (fBM) with Hurst parameter $H\in(1/4,1)$ in terms of Newtonian-type capacities that are defined with respect to the (sub-Riemannian) control distance associated with the vector fields. As a starting point, we first establish the existence and smoothness of joint densities for the finite-dimensional distributions of the solution in the general context of hypoellitpic SDEs driven by Gaussian rough paths. We then turn to the fBM setting and derive a local upper bound for the joint density in terms of the control distance. As an application of these results, we establish our main estimate on hitting probabilities which generalises a well-known elliptic result of \cite{BNOT} to the hypoelliptic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hitting Probabilities for Hypoelliptic Differential Equations Driven by Fractional Brownian Motion Geng, Xi Wang, Sheng Probability The main goal of this article is to derive a two-sided estimate for hitting probabilities of a hypoelliptic stochastic differential equation (SDE) driven by fractional Brownian motion (fBM) with Hurst parameter $H\in(1/4,1)$ in terms of Newtonian-type capacities that are defined with respect to the (sub-Riemannian) control distance associated with the vector fields. As a starting point, we first establish the existence and smoothness of joint densities for the finite-dimensional distributions of the solution in the general context of hypoellitpic SDEs driven by Gaussian rough paths. We then turn to the fBM setting and derive a local upper bound for the joint density in terms of the control distance. As an application of these results, we establish our main estimate on hitting probabilities which generalises a well-known elliptic result of \cite{BNOT} to the hypoelliptic case. |
| title | Hitting Probabilities for Hypoelliptic Differential Equations Driven by Fractional Brownian Motion |
| topic | Probability |
| url | https://arxiv.org/abs/2512.07202 |