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

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.