Abstract
Let \(\Phi \) be a nuclear space, and let \(\Phi '\) denote its strong dual. In this paper, we introduce sufficient conditions for the almost sure uniform convergence on bounded intervals of time for a sequence of \(\Phi '\)-valued processes having continuous (respectively, càdlàg) paths. The main result is formulated first in the general setting of cylindrical processes but later specialized to other situations of interest. In particular, we establish conditions for the convergence to occur in a Hilbert space continuously embedded in \(\Phi '\). Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to convergence in \(L^{r}\) uniformly on a bounded interval of time, to the convergence of a series of independent càdlàg processes, and to the convergence of solutions to linear stochastic evolution equations driven by Lévy noise.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, Cambridge (2009)
Basse-O’Connor, A., Rosiński, J.: On the uniform convergence of random series in Skorohod space and representations of càdlàg infinitely divisible processes. Ann. Probab. 41(6), 4317–4341 (2013)
Bogachev, V.I.: Measure Theory, vol. 1. Springer, Berlin (2007)
Dalecky, Y.L., Fomin, S.V.: Measure and Differential Equations in Infinite-Dimensional Space. Mathematics and its Applications, Springer, Berlin (1991)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Series in Advanced Mathematics, Cambridge University Press, Cambridge (1995)
Fonseca-Mora, C.A.: Existence of continuous and Càdlàg versions for cylindrical processes in the dual of a nuclear space. J. Theor. Probab. 31(2), 867–894 (2018)
Fonseca-Mora, C.A.: Stochastic integration and stochastic PDEs driven by jumps on the dual of a nuclear space. Stoch. PDE Anal. Comput. 6(4), 618–689 (2018)
Fonseca-Mora, C.A.: Lévy processes and infinitely divisible measures in the dual of a nuclear space. J. Theor. Probab. 33(2), 649–691 (2020)
Fonseca-Mora, C.A.: Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications. Studia Math. 254(2), 109–147 (2020)
Fonseca-Mora, C.A.: Stochastic evolution equations with Lévy noise in the dual of a nuclear space. Stoch. PDE Anal. Comput. (2022). https://doi.org/10.1007/s40072-022-00281-7
Jarchow, H.: Locally Convex Spaces. Mathematische Leitfäden, Springer, Berlin (1981)
Kallianpur, G., Pérez-Abreu, V.: Stochastic evolution equations driven by nuclear-space-valued martingales. Appl. Math. Optim. 17, 125–172 (1988)
Kallianpur, G., Pérez-Abreu, V.: Weak Convergence of Solutions of Stochastic Evolution Equations on Nuclear spaces. Stochastic partial differential equations and applications II (Trento, 1988). Lecture Notes in Math., vol. 1390, pp. 119–132. Springer, Berlin (1989)
Kumar, U., Riedle, M.: The stochastic Cauchy problem driven by a cylindrical Lévy process. Electron. J. Probab. 25(10), 26 (2020)
Mitoma, I.: On the sample continuity of \(\mathscr {S}^{\prime }\)-processes. J. Math. Soc. Jpn 35(4), 629–636 (1983)
Mitoma, I.: Almost Sure Uniform Convergence of Continuous Stochastic Processes with Values in the Dual of a Nuclear Space. Probability theory and mathematical statistics (Tbilisi, 1982). Lecture Notes in Math., pp. 446–451. Springer, Berlin (1983)
Narici, L., Beckenstein, E.: Topological Vector Spaces. Pure and Applied Mathematics, 2nd edn. CRC Press, Boca Raton (2011)
Pérez-Abreu, V., Tudor, C.: Regularity and convergence of stochastic convolutions in duals of nuclear Fréchet spaces. J. Multivariate Anal. 43(2), 185–199 (1992)
Pietsch, A.: Nuclear Locally Convex Spaces. Ergebnisse der Mathematikund ihrer Grenzgebiete, Springer, Berlin (1972)
Schaefer, H.: Topological Vector Spaces. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1999)
Sun, X., Xie, L., Xie, Y.: Pathwise uniqueness for a class of SPDEs driven by cylindrical \(\alpha \)-stable processes. Potential Anal. 53(2), 659–675 (2020)
Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Pure and Applied Mathematics, Academic Press, Cambridge (1967)
van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255(4), 940–993 (2008)
Veraar, M., Yaroslavtsev, I.: Cylindrical continuous martingales and stochastic integration in infinite dimensions. Electron. J. Probab. 21(59), 1–53 (2016)
Wu, J.-L.: On the regularity of stochastic difference equations in hyperfinite-dimensional vector spaces and applications to \({\mathscr {D}}^{\prime }\)-valued stochastic differential equations. Proc. R. Soc. Edinb. Sect. A 124(6), 1089–1117 (1994)
Acknowledgements
This work was supported by The University of Costa Rica through the grant “821-C2-132- Procesos cilíndricos y ecuaciones diferenciales estocásticas.” The author thanks two anonymous referees for valuable comments and suggestions that contributed greatly to improve the presentation of this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Fonseca-Mora, C.A. Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space. J Theor Probab 36, 2564–2589 (2023). https://doi.org/10.1007/s10959-023-01243-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-023-01243-y
Keywords
- Cylindrical stochastic processes
- Processes with continuous and càdlàg paths
- Almost sure uniform convergence
- Dual of a nuclear space