L\'evy measures on Banach spaces

TL;DR

This paper provides explicit characterizations of Lévy measures on $L^p$-spaces and UMD Banach spaces, extending known descriptions and introducing new integrability conditions for infinite-dimensional Banach spaces.

Contribution

It offers the first explicit characterizations of Lévy measures on UMD Banach spaces, generalizing previous results on sequence spaces and $L^p$-spaces.

Findings

Lévy measures on $L^p$-spaces are characterized by an integrability condition.

Lévy measures on UMD Banach spaces are characterized by the finiteness of a $ extgamma$-radonifying norm expectation.

The description simplifies to integrability conditions in the case of $L^p$-spaces.

Abstract

We establish an explicit characterisation of L\'evy measures on both $L^p$-spaces and UMD Banach spaces. In the case of $L^p$-spaces, L\'evy measures are characterised by an integrability condition, which directly generalises the known description of L\'evy measures on sequence spaces. The latter has been the only known description of L\'evy measures on infinite dimensional Banach spaces that are not Hilbert. L\'evy measures on UMD Banach spaces are characterised by the finiteness of the expectation of a random {\gamma}-radonifying norm. Although this description is more abstract, it reduces to simple integrability conditions in the case of $L^p$-spaces.