Tightness and weak convergence in the topology of local uniform convergence for stochastic processes in the dual of a nuclear space

TL;DR

This paper establishes conditions for tightness and weak convergence of probability measures on the space of $ ext{local uniform}$ continuous $ ext{dual nuclear space}$-valued processes, with applications to martingales and SPDEs.

Contribution

It provides new sufficient conditions for tightness and weak convergence in the topology of local uniform convergence for processes in the dual of a nuclear space, with practical applications.

Findings

Proved tightness criteria for probability measures on $C_{ abla}( ext{dual nuclear space})$.

Established weak convergence results for sequences of $ ext{dual nuclear space}$-valued processes.

Applied results to central limit theorem for local martingales and solutions to stochastic PDEs.

Abstract

Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$ and let $C_{\infty}(\Phi')$ be the collection of all continuous mappings $x:[0,\infty) \rightarrow \Phi'$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty}(\Phi')$ and for weak convergence in $C_{\infty}(\Phi')$ for a sequence of $\Phi'$-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 $C_{\infty}(\Phi')$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.