{
  "abstract": "The goal of the present paper is to investigate the exact decay rate of the tail probability $\\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assumption of uniform ellipticity, we establish a general $(2H+1)$-Weibull lower tail estimate in Young's regime of $H\\in(1/2,1)$ and a general Gaussian lower tail estimate in the rough regime of $H\\in(1/4,1/2)$. These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with $C_b^\\infty$-coefficients could \\textit{fail} to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \\cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any $2\\times 2$ periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general $(2H+1)$-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \\cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06152",
  "author_total": 3,
  "authors": [
    {
      "name": "Horatio Boedihardjo"
    },
    {
      "name": "Xi Geng"
    },
    {
      "name": "Sheng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dnvg2jycss15q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06152",
  "fulltext_url": null,
  "id": "dnvg2jycss15q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06152",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7220677406",
  "pdf_url": "https://arxiv.org/pdf/2610.06152",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dnvg2jycss15q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06152",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06152"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion",
  "title_text": "Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dnvg2jycss15q",
  "venue": "arXiv",
  "versions": []
}