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

TL;DR

This paper establishes sufficient conditions for almost sure uniform convergence of sequences of nuclear space dual-valued processes, with applications to series of Lévy-driven processes and solutions to linear evolution equations.

Contribution

It introduces new criteria for uniform convergence of nuclear space dual processes, including specialized results for Hilbert space embeddings and ultrabornological nuclear spaces.

Findings

Conditions for convergence in Hilbert space embedded in the dual.

Applications to series of Lévy-driven processes.

Convergence results for solutions to linear evolution equations.

Abstract

Let $\Phi$ be a nuclear space and let $\Phi'$ 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 $\Phi'$-valued processes having continuous (respectively c\`{a}dl\`{a}g) 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 the convergence of a series of independent c\`{a}dl\`{a}g process and to the convergence of solutions to linear evolution equations driven by L\'{e}vy noise.