{
  "abstract": "We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\\int_0^t S(t-s)g(s)\\,\\mathrm{d}β_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\\in L^\\infty(Ω;L^2(0,T;X))$, where $β$ is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator $A$ has sectorial angle $π/2$ and a bounded $H^\\infty$-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an $L^2$-maximal estimate on $\\mathbb R_+$ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary $C_0$-semigroups under the stronger condition $g\\in L^2([0,T]; L^\\infty(Ω;X))$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12688",
  "author_total": 2,
  "authors": [
    {
      "name": "Mark Veraar"
    },
    {
      "name": "Joris van Winden"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24b1vkpqqn9z9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12688",
  "fulltext_url": null,
  "id": "24b1vkpqqn9z9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12688",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212810015",
  "pdf_url": "https://arxiv.org/pdf/2609.12688",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24b1vkpqqn9z9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12688",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12688"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "No free lunch for continuity of stochastic convolutions",
  "title_text": "No free lunch for continuity of stochastic convolutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/24b1vkpqqn9z9",
  "venue": "arXiv",
  "versions": []
}