{
  "abstract": "Let $B$ be a standard Brownian motion, $x\\ge 0,\\, ν<1$, and $b:[0,\\infty)\\to\\mathbb R$ is a continuous function locally of finite variation starting from 0. We define the perturbed Brownian motion reflected at the boundary $b$ by establishing strong existence and pathwise uniqueness of a solution to the equation \\[ W_t=(1-ν)x+B_t+νM_t(W)+\\frac12 L_t^0(W-b), \\qquad W_t\\ge b(t), \\] where $M_t(W):=\\sup_{0\\le s\\le t} W_s$ and the process $L^0(W-b)$ is the semimartingale local time at 0 of the process $W-b$. We give a positive result under condition (PB) on the boundary $b$ : for every $T>0$, the upward increment $\\sup_{0\\le s<t\\le T,\\,t-s\\le h}(b(t)-b(s))^+ = o(\\sqrt{h})$ as $h\\downarrow 0$. The proof splits into two regimes: the case $ν<1/2$ is a consequence of the Skorokhod problem in an orthant proved by [Williams 1995], while the case $ν\\ge 1/2$ combines a deterministic comparison estimate and a logarithmic upper bound on the number of completed round-trips, following the strategy of [Chaumont and Doney 1999]. For $α\\in(0,1/2)$, we also construct an increasing $α$-Hölder boundary for which no continuous adapted solution starting from zero exists for any $ν<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20491",
  "author_total": 1,
  "authors": [
    {
      "name": "Chengshi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26fr6w8vpgw9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20491",
  "fulltext_url": null,
  "id": "26fr6w8vpgw9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20491",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7213604763",
  "pdf_url": "https://arxiv.org/pdf/2609.20491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26fr6w8vpgw9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Perturbed Brownian motion reflected at a time-dependent boundary",
  "title_text": "Perturbed Brownian motion reflected at a time-dependent boundary",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/26fr6w8vpgw9c",
  "venue": "arXiv",
  "versions": []
}