Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

Let $$\Phi $$Φ be a nuclear space and let $$\Phi '_{\beta }$$Φβ′ denote its strong dual. In this work, we prove the existence of càdlàg versions, the Lévy–Itô decomposition and the Lévy–Khintchine formula for $$\Phi '_{\beta }$$Φβ′-valued Lévy processes. Moreover, we give a characterization for Lévy measures on $$\Phi '_{\beta }$$Φβ′ and provide conditions for the existence of regular versions to cylindrical Lévy processes in $$\Phi '$$Φ′. Furthermore, under the assumption that $$\Phi $$Φ is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on $$\Phi '_{\beta }$$Φβ′ and Lévy processes in $$\Phi '_{\beta }$$Φβ′. Finally, we prove the Lévy–Khintchine formula for infinitely divisible measures on $$\Phi '_{\beta }$$Φβ′.