On infinitely divisible laws in $C[0,1]$

Abstract:

In Euclidean spaces, or in a separable Hilbert space, the characteristic function of an infinitely divisible distribution has the familiar form given by the Lévy-Khintchine formula. The Lévy measures $M$ of this formula are characterized by the property that the integral of $\min [1,||x|{|^2}]$ with respect to $M$ is finite. This simple situation no longer holds in the Banach space $C = C[0,1]$ where integrability of $\min [1,||x||]$ is sufficient but integrability of $\min [1,||x|{|^2}]$ is neither necessary nor sufficient. Certain other conditions which are sufficient to imply that $M$ is the Lévy measure of a distribution on $C$ can be obtained with the use of an integral formula of Garsia.