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

Let Phi be a nuclear space and let Phi'(beta) denote its strong dual. In this work, we prove the existence of cadlag versions, the Levy-Ito decomposition and the Levy-Khintchine formula for Phi'(beta)-valued Levy processes. Moreover, we give a characterization for Levy measures on Phi'(beta) and provide conditions for the existence of regular versions to cylindrical Levy processes in Phi'. Furthermore, under the assumption that Phi is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on Phi'(beta) and Levy processes in Phi'(beta). Finally, we prove the Levy-Khintchine formula for infinitely divisible measures on Phi'(beta).