# L\'{e}vy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

## TL;DR

This paper develops a comprehensive theoretical framework linking Lévy processes and infinitely divisible measures in the dual of a nuclear space, including key formulas and decompositions, advancing the mathematical understanding of infinite-dimensional stochastic processes.

## Contribution

It establishes a one-to-one correspondence between Lévy processes and infinitely divisible measures in the dual of a nuclear space, including Lévy-Itô decomposition and Lévy-Khintchine formula.

## Key findings

- Proves the Lévy-Itô decomposition for Lévy processes in the dual space.
- Derives the Lévy-Khintchine formula for infinitely divisible measures.
- Characterizes Lévy measures on the dual of a nuclear space.

## Abstract

Let $\Phi$ be a nuclear space and let $\Phi'_{\beta}$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $\Phi'_{\beta}$ and L\'{e}vy processes taking values in $\Phi'_{\beta}$. Moreover, we prove the L\'{e}vy-It\^{o} decomposition, the L\'{e}vy-Khintchine formula and the existence of c\`{a}dl\`{a}g versions for $\Phi'_{\beta}$-valued L\'{e}vy processes. A characterization for L\'{e}vy measures on $\Phi'_{\beta}$ is also established. Finally, we prove the L\'{e}vy-Khintchine formula for infinitely divisible measures on $\Phi'_{\beta}$.