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

Abstract

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

To appear in Journal of Theoretical Probability

References in corpus (2)

• Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space
• Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space

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• Almost Sure Uniform Convergence of Stochastic Processes in the Dual of a Nuclear Space
• Time regularity of stochastic convolutions and stochastic evolution equations in duals of nuclear spaces
• Convergence Uniform on Compacts in Probability with Applications to Stochastic Analysis in Duals of Nuclear Spaces