Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space

Abstract

Let $\Phi$ be a nuclear space and let ${\Phi }^{\prime }$ denote its strong dual. In this paper we introduce sufficient conditions for the almost surely uniform convergence on bounded intervals of time for a sequence of ${\Phi }^{\prime }$-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 ${\Phi }^{\prime }$. Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to the convergence of a series of independent càdlàg process and to the convergence of solutions to linear evolution equations driven by Lévy noise.

References in corpus (5)

• Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space
• Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space
• Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space
• Tightness and Weak Convergence of Probabilities on the Skorokhod Space on the Dual of a Nuclear Space and Applications
• Stochastic Evolution Equations with Lévy Noise in the Dual of a Nuclear Space

Cited by in corpus (1)

• Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces