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Weak convergence for continuous stochastic processes in the dual of a nuclear space with applications to the convergence of SPDEs

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Veröffentlicht/Copyright: 12. Juni 2025
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Random Operators and Stochastic Equations
Aus der Zeitschrift Random Operators and Stochastic Equations Band 33 Heft 4

Abstract

Let Φ ′ denote the strong dual of a nuclear space Φ and let C ∞ ⁢ ( Φ ′ ) be the collection of all continuous mappings x : [ 0 , ∞ ) → Φ ′ equipped with the topology of local uniform convergence. In this paper, we prove sufficient conditions for tightness of probability measures on C ∞ ⁢ ( Φ ′ ) and for weak convergence in C ∞ ⁢ ( Φ ′ ) for a sequence of Φ ′ -valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in C ∞ ⁢ ( Φ ′ ) for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

Keywords: Local uniform convergence; uniform tightness; weak convergence; central limit theorem; dual of a nuclear space
MSC 2020: 60B10; 60B12; 60G48; 60H15

Funding source: Universidad de Costa Rica

Award Identifier / Grant number: 821-C2-132

Funding statement: This work was supported by The University of Costa Rica through the grant “821-C2-132 – Procesos cilíndricos y ecuaciones diferenciales estocásticas”.

Acknowledgements

The author would like to thank Adrián Barquero-Sánchez for helpful suggestions that contributed greatly to improve the presentation of this article.

  1. Communicated by: Vyacheslav L. Girko

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Received: 2024-09-11
Accepted: 2025-04-08
Published Online: 2025-06-12
Published in Print: 2025-11-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Forward-backward doubly stochastic differential equations with Poisson jumps in infinite dimensions
  3. The global elliptic law for complex random matrices
  4. Estimation for a misspecified change point type signal driven by a fractional Brownian motion
  5. Averaging result for impulsive 𝜓-Hilfer fractional stochastic pantograph-type delay system driven by Poisson jumps
  6. On a switching control problem with làdlàg costs
  7. Quasi-(𝑚,𝑛)-paranormal operators
  8. Weak convergence for continuous stochastic processes in the dual of a nuclear space with applications to the convergence of SPDEs
https://doi.org/10.1515/rose-2025-2026
Schlagwörter für diesen Artikel
Local uniform convergence; uniform tightness; weak convergence; central limit theorem; dual of a nuclear space

Artikel in diesem Heft

  1. Frontmatter
  2. Forward-backward doubly stochastic differential equations with Poisson jumps in infinite dimensions
  3. The global elliptic law for complex random matrices
  4. Estimation for a misspecified change point type signal driven by a fractional Brownian motion
  5. Averaging result for impulsive 𝜓-Hilfer fractional stochastic pantograph-type delay system driven by Poisson jumps
  6. On a switching control problem with làdlàg costs
  7. Quasi-(𝑚,𝑛)-paranormal operators
  8. Weak convergence for continuous stochastic processes in the dual of a nuclear space with applications to the convergence of SPDEs
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