bibkey: dzhenzhersakbaev2026slln
authors: Sviatoslav V. Dzhenzher; Vsevolod Zh. Sakbaev
year: 2026
title: "The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces"
doi: 10.1007/s43036-025-00481-7
url: https://arxiv.org/abs/2507.07658v3
claim: "The paper proves a strong law of large numbers for products of exponentials of i.i.d. bounded random operators: on a uniformly smooth Banach space the products exp(A_1 t/n)...exp(A_n t/n) converge almost surely in the strong operator topology to exp(t E A), uniformly for t in any segment; it proves the weak-operator version on every Banach space, deduces the strong version on l_1 by Schur's theorem, and conjectures the strong version on every Banach space."
strata_touched:
  - D5/S3/Analytic/Probability/RandomExponentialProductSLLN
license: citation-only
triage: anchor

Dzhenzher and Sakbaev, the strong law of large numbers for random semigroups

S. V. Dzhenzher and V. Zh. Sakbaev, arXiv:2507.07658v3 (18 October 2025; v1 10 July 2025); Advances in Operator Theory 11 (2026), no. 1, article 4. Quotations are from the v3 TeX source, with macros expanded and theorem references replaced by bracketed descriptions.

Setting (Section 2): random elements of a Banach space are Bochner measurable maps on a complete probability space, random operators are random elements of the bounded operators $\mathcal{L}(\mathcal{X})$ with the operator norm, and expectations are Bochner integrals.

Theorem 2.4 (SLLN in SOT):

Let $\mathcal{X}$ be a uniformly smooth Banach space. Let $A:\Omega \to \mathcal{L}(\mathcal{X})$ be a generator of a semigroup $e^{At}$ such that $\Vert A\Vert$ is bounded. Let $A,A_1,A_2,\dots$ be a sequence of i.i.d. random generators. Then $e^{A_{1}t/n}\dots e^{A_{n}t/n}$ converges a.s. in SOT of $\Vert \cdot {\Vert }_{\mathcal{X}}$ to $e^{\mathbb{E}At}$ uniformly for $t>0$ in any segment

that is, for every $x$ and $T>0$, almost surely ${\lim }_n{\sup }_{t\in [0,T]}\Vert (e^{A_{1}t/n}\cdots e^{A_{n}t/n}-e^{\mathbb{E}At})x\Vert =0$. The weak-operator version holds on every Banach space (Theorem 2.5), which gives the strong version on ${\ell }_1$, a space that is not uniformly smooth.

Conjecture 2.6:

Let $\mathcal{X}$ be a Banach space. Then under the assumptions of [Theorem 2.4], $e^{A_{1}t/n}\dots e^{A_{n}t/n}$ converges a.s. in SOT of $\Vert \cdot {\Vert }_{\mathcal{X}}$ to $e^{\mathbb{E}At}$ uniformly for $t>0$ in any segment.

Verified locator

Source: https://arxiv.org/abs/2507.07658v3

• DOI: https://doi.org/10.1007/s43036-025-00481-7 (Advances in Operator Theory 11 (2026), no. 1, article 4; author list, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/2507.07658v3 (the e-print is a gzipped tar archive; the main text is the member mains/slln_smooth.tex, 603 lines, md5 a977c94eb4adc6b7a26664cb45150780, included by main_arxiv.tex):
  • random elements, random operators and expectations, l. 57–193;
  • Theorem 2.4, l. 195–204;
  • the conjecture, l. 224–229.
• The arXiv version dates are from the arXiv metadata.