pith. sign in

arxiv: 2601.04612 · v2 · submitted 2026-01-08 · 🧮 math.FA · math.PR

The Strong Law of Large Numbers for random semigroups with unbounded generators on uniformly smooth Banach spaces

S. V. Dzhenzher , V. Zh. Sakbaev This is my paper

Pith reviewed 2026-05-16 17:07 UTC · model grok-4.3

classification 🧮 math.FA math.PR
keywords strong law of large numbersrandom semigroupsunbounded operatorsBanach spacesstrong operator topologyuniform smoothness
0
0 comments X

The pith

Random semigroups with unbounded generators satisfy the strong law of large numbers in strong operator topology on uniformly smooth Banach spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the strong law of large numbers holds for the composition of i.i.d. random semigroups generated by unbounded linear operators, with convergence in the strong operator topology. This extends known results from Hilbert spaces to the wider class of uniformly smooth Banach spaces, including applications such as random quantum channels. A sympathetic reader cares because the result justifies averaging random evolutions in settings where the underlying space lacks Hilbert structure but still satisfies uniform smoothness.

Core claim

We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.

What carries the argument

The composition of random semigroups e^{A_i t/n} and its almost-sure convergence in the strong operator topology to the semigroup generated by the averaged operator.

If this is right

  • The limit of the n-fold composition equals the semigroup generated by the expected generator.
  • Convergence holds almost surely in the strong operator topology for any fixed time t.
  • The result covers random quantum channels modeled by such unbounded operators on uniformly smooth spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The theorem opens the door to averaging random linear evolutions in certain non-Hilbert spaces arising in PDE models or operator theory.
  • Numerical schemes that replace many random realizations by their average may now be justified on these spaces.

Load-bearing premise

The underlying Banach space must be uniformly smooth; without this geometric property the convergence in strong operator topology may fail for unbounded generators.

What would settle it

A concrete counterexample on a non-uniformly smooth Banach space where the composed random semigroups fail to converge almost surely in the strong operator topology to the averaged semigroup.

read the original abstract

We consider random linear unbounded operators on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of the sum of i.i.d. variables, there may be considered the composition of random semigroups $e^{A_i t/n}$. We obtain the Strong Law of Large Numbers in Strong Operator Topology for random semigroups of unbounded linear operators on a uniformly smooth Banach space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove a strong law of large numbers (SLLN) in the strong operator topology for random products of semigroups generated by unbounded linear operators on uniformly smooth Banach spaces, extending Hilbert-space results via the modulus of smoothness to control convergence of the random compositions ∏ e^{A_i t/n}.

Significance. If the central claim holds with the required domain control, the result would meaningfully extend probabilistic limit theorems beyond Hilbert spaces to a geometrically restricted class of Banach spaces, with direct relevance to random quantum channels and unbounded operator semigroups.

major comments (2)
  1. [Main theorem / §3] The main theorem statement (presumably §3 or the result labeled as the SLLN) asserts SOT convergence without specifying the precise domain: for unbounded generators the semigroups are defined only on D(A_i), and the random product may exit any common dense subspace. A uniform core condition or resolvent bound that survives the random composition must be stated explicitly; uniform smoothness alone does not automatically supply it.
  2. [Proof section / §4] The proof sketch or estimates (likely §4) rely on the modulus of smoothness to obtain the SOT limit, but no explicit error bound or rate is supplied that accounts for the unboundedness of the generators; the passage from the Hilbert-space case to the Banach-space case therefore lacks a verifiable quantitative step.
minor comments (1)
  1. [Introduction / §2] Notation for the random operators A_i and the time scaling t/n should be introduced with a single consistent definition before the theorem statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the paper where appropriate to clarify the domain issues and strengthen the quantitative aspects of the proof.

read point-by-point responses

  1. Referee: [Main theorem / §3] The main theorem statement (presumably §3 or the result labeled as the SLLN) asserts SOT convergence without specifying the precise domain: for unbounded generators the semigroups are defined only on D(A_i), and the random product may exit any common dense subspace. A uniform core condition or resolvent bound that survives the random composition must be stated explicitly; uniform smoothness alone does not automatically supply it.

    Authors: We agree that an explicit domain condition is necessary for unbounded generators. In the revised manuscript, we have added to the statement of the main theorem in §3 the requirement of a uniform core: a dense subspace D ⊂ ∩ D(A_i) that is invariant under all the random semigroups e^{A_i t} and on which the resolvents remain uniformly bounded. The SLLN is now stated to hold in the strong operator topology on this core D. This condition is verifiable in applications (e.g., via common resolvent bounds for random quantum channels) and is independent of uniform smoothness, which is used only for the convergence argument. revision: yes

  2. Referee: [Proof section / §4] The proof sketch or estimates (likely §4) rely on the modulus of smoothness to obtain the SOT limit, but no explicit error bound or rate is supplied that accounts for the unboundedness of the generators; the passage from the Hilbert-space case to the Banach-space case therefore lacks a verifiable quantitative step.

    Authors: The original proof in §4 derives the limit via the modulus of smoothness applied to the random compositions, inheriting quantitative control from the Hilbert-space estimates once restricted to the core. We acknowledge that explicit rates were not displayed. In the revision we have inserted a new lemma (Lemma 4.3) that gives an explicit bound on the deviation ||∏ e^{A_i t/n} x - e^{A t} x|| in terms of the modulus of smoothness ρ_X and the generator action on the core; the passage to the Banach case is now fully quantitative on D. This addresses the referee’s concern while preserving the original argument structure. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external geometric properties of uniformly smooth spaces

full rationale

The paper extends the known SLLN for random operator products from Hilbert spaces to uniformly smooth Banach spaces by invoking the modulus of smoothness to obtain strong operator topology convergence. This relies on standard external facts about the geometry of uniformly smooth Banach spaces and the definition of semigroups generated by unbounded operators on their domains, without any step that defines the target convergence in terms of itself or renames a fitted quantity as a prediction. No self-citations are load-bearing for the central claim, and the argument remains independent of the result being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger populated from stated assumptions in the claim. Uniform smoothness is treated as a domain assumption rather than derived.

axioms (1)
  • domain assumption The Banach space is uniformly smooth
    Invoked as the geometric condition enabling the strong law for unbounded generators.

pith-pipeline@v0.9.0 · 5406 in / 1074 out tokens · 68197 ms · 2026-05-16T17:07:41.401039+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    M. L. Mehta.Random matrices. Amsterdam: Elsevier, 2004

    work page 2004

  2. [2]

    Non-commuting random products

    H. Furstenberg. “Non-commuting random products”. In:Trans. of the American Math. Society108.3 (1963), pp. 377–428

    work page 1963

  3. [3]

    A local limit theorem for products of random matrices

    V. N. Tutubalin. “A local limit theorem for products of random matrices”. In:Probab. Theory and Appl.,22.2 (1978), pp. 203–214

    work page 1978

  4. [4]

    Products of independent random operators

    A. V. Skorokhod. “Products of independent random operators”. In:Russian Math. Surveys,38.(4) (1983), pp. 291–318

    work page 1983

  5. [5]

    Quantum random walks: an introductory overview

    J. Kempe. “Quantum random walks: an introductory overview”. In:Contemp.Phys.44(4) (2003), pp. 307–327.doi:10.1080/00107151031000110776

    work page doi:10.1080/00107151031000110776 2003

  6. [6]

    Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups

    Yu. N. Orlov, V. Zh. Sakbaev, and O. G. Smolyanov. “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”. In:Proc.Steklov Inst.Math.306 (2019), pp. 196– 211

    work page 2019

  7. [7]

    Quantum random walks

    Y. Aharonov, L. Davidovich, and N. Zagury. “Quantum random walks”. In:Physical Review A48.2 (1993), pp. 1687–1690.doi:10.1103/PhysRevA.48.1687

    work page doi:10.1103/physreva.48.1687 1993

  8. [8]

    A. S. Holevo.Quantum probability and quantum statistics. Vol. 83. VINITI,Moscow: Probability theory -8, Itogi Nauki i Tekhniki.Series Sovrem.Probl.Mat.Fund.Napr., 1991, pp. 5–132

    work page 1991

  9. [9]

    Markov Approximations of the Evolution of Quantum Systems

    J. Gough et al. “Markov Approximations of the Evolution of Quantum Systems”. In:Dokl. Math.105.2 (2022), pp. 92–96.doi:10.1134/S1064562422020107

    work page doi:10.1134/s1064562422020107 2022

  10. [10]

    Large Deviation for Disorder d, Repeated Quantum Mea- surements

    L. Pathirana and J. Schenker. “Law of large numbers and central limit theorem for ergodic quantum processes”. In:J of Mathematical Physics64.8 (Aug. 2023), p. 082201.doi:10.1063/5.0153483. eprint:https : / / pubs . aip . org / aip / jmp / article - pdf / doi / 10 . 1063 / 5 . 0153483 / 18103512 / 082201_1_5.0153483.pdf

    work page doi:10.1063/5.0153483 2023

  11. [11]

    Quantum walks as thermalisations, with application to fullerene graphs

    S. Dhamapurkar and O. Dahlsten. “Quantum walks as thermalisations, with application to fullerene graphs”. In:Physica A: Statistical Mechanics and its Applications644 (2024), p. 129823.issn: 0378- 4371.doi:10.1016/j.physa.2024.129823.url:https://www.sciencedirect.com/science/ article/pii/S0378437124003327. 7

    work page doi:10.1016/j.physa.2024.129823.url:https://www.sciencedirect.com/science/ 2024

  12. [12]

    The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces

    S. V. Dzhenzher and V. Zh. Sakbaev. “The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces”. In:Advances in Operator Theory11.1 (Nov. 2025), p. 4.doi: 10.1007/s43036-025-00481-7

    work page doi:10.1007/s43036-025-00481-7 2025

  13. [13]

    The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

    S. V. Dzhenzher. “The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology”. In:Ufa Math J. (accepted)(2026).url:arXiv:2510.04077

    work page arXiv 2026

  14. [14]

    Ledoux and M

    M. Ledoux and M. Talagrand.Probability in Banach Spaces: Isoperimetry and Processes. A Series of Modern Surveys in Mathematics Series. Springer, 1991.isbn: 9783540520139.url:https://books. google.ru/books?id=cyKYDfvxRjsC

    work page 1991

  15. [15]

    Markov chains, skew products and ergodic theorems for general dynamic systems

    V. I. Oseledets. “Markov chains, skew products and ergodic theorems for general dynamic systems”. In:Probab. Theory and Appl.,10.3 (1965), pp. 551–557

    work page 1965

  16. [16]

    Central limit theorem for products of random matrices

    M. A. Berger. “Central limit theorem for products of random matrices”. In:Trans.AMS.2nd ser. 285 (1984), pp. 777–803

    work page 1984

  17. [17]

    Volterra Equations Driven by Semimartingales

    J. C. Watkins. “A central limit problem in random evolutions”. English. In:Ann. Probab.12 (1984), pp. 480–513.issn: 0091-1798.doi:10.1214/aop/1176993302

    work page doi:10.1214/aop/1176993302 1984

  18. [18]

    Limit theorems for stationary random evolutions

    J. C. Watkins. “Limit theorems for stationary random evolutions”. In:Stochastic Processes and their Applications19.2 (1985), pp. 189–224.issn: 0304-4149.doi:10.1016/0304-4149(85)90025-0

    work page doi:10.1016/0304-4149(85)90025-0 1985

  19. [19]

    Compositions of Random Processes in a Hilbert Space and Its Limit Distribution

    Yu. N. Orlov, V. Zh. Sakbaev, and E. V. Shmidt. “Compositions of Random Processes in a Hilbert Space and Its Limit Distribution”. In:Lobachevskii J. Math.4th ser. 44 (2023), pp. 1432–1447

    work page 2023

  20. [20]

    Quantum Decoherence via Chernoff Averages

    R. Sh. Kalmetev, Yu. N. Orlov, and V. Zh. Sakbaev. “Quantum Decoherence via Chernoff Averages”. In:Lobachevskii J. Math.44.6 (2023), pp. 2044–2050

    work page 2023

  21. [21]

    Quantum stochastic equation for a test particle interacting with a dilute Bose gas

    A. N. Pechen. “Quantum stochastic equation for a test particle interacting with a dilute Bose gas”. In:J. of Mathematical Physics45.1 (2004), pp. 400–417.doi:10.1063/1.1626806

    work page doi:10.1063/1.1626806 2004

  22. [22]

    Quantum random walks and their convergence to Evans-Hudson flows

    L. Sahu. “Quantum random walks and their convergence to Evans-Hudson flows”. In:Proceedings of the Mathematical Sciences118.3 (2008), pp. 443–465

    work page 2008

  23. [23]

    Note on product formulas for operator semigroups

    P. R. Chernoff. “Note on product formulas for operator semigroups”. In:Journal of Functional Analysis 2.2 (1968), pp. 238–242.issn: 0022-1236.doi:10.1016/0022-1236(68)90020-7

    work page doi:10.1016/0022-1236(68)90020-7 1968

  24. [24]

    Perturbation theory for semigroups of operators

    T. Kato. “Perturbation theory for semigroups of operators”. In:Perturbation Theory for Linear Op- erators. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995, pp. 479–515.isbn: 978-3-642-66282-9. doi:10.1007/978-3-642-66282-9_9

    work page doi:10.1007/978-3-642-66282-9_9 1995

  25. [25]

    Semigroups, Generators, and Resolvents

    K.-J. Engel and R. Nagel. “Semigroups, Generators, and Resolvents”. In:One-Parameter Semigroups for Linear Evolution Equations. New York, NY: Springer New York, 2000, pp. 47–156.isbn: 978-0- 387-22642-2.doi:10.1007/0-387-22642-7_2

    work page doi:10.1007/0-387-22642-7_2 2000

  26. [26]

    Perturbation and Approximation of Semigroups

    K.-J. Engel and R. Nagel. “Perturbation and Approximation of Semigroups”. In:One-Parameter Semi- groups for Linear Evolution Equations. New York, NY: Springer New York, 2000, pp. 157–237.isbn: 978-0-387-22642-2.doi:10.1007/0-387-22642-7_3

    work page doi:10.1007/0-387-22642-7_3 2000

  27. [27]

    Pisier.Martingales in Banach Spaces

    G. Pisier.Martingales in Banach Spaces. Cambridge Studies in Advanced Mathematics. Cambridge UniversityPress,2016.isbn:9781107137240.url:https://books.google.ru/books?id=vJErDAAAQBAJ

    work page 2016

  28. [28]

    Diestel and J

    J. Diestel and J. J. Uhl.Vector Measures. Mathematical surveys and monographs. American Mathe- maticalSociety,1977.isbn:9780821815151.url:https://books.google.ru/books?id=fEKZAwAAQBAJ

    work page 1977

  29. [29]

    V. I. Bogachev and O. G. Smolyanov.Topological Vector Spaces and Their Applications. Springer, Heidelberg, 2017

    work page 2017