Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space (1701.06630v2)

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 $\Phi'{\beta}$.