On a family of Levy processes without support in S'

TL;DR

This paper investigates a specific family of Levy processes with support properties in distribution spaces, which is crucial for advanced stochastic modeling and solving stochastic partial differential equations.

Contribution

It characterizes the support of a family of Levy processes in various distribution spaces, expanding understanding of their structure and applications.

Findings

Levy processes with 1<α<2 are supported in K' (distributions of exponential type)

Levy processes with 0<α≤1 are supported in similar spaces of power exponential type

The support properties are significant for stochastic modeling and SPDE solutions.

Abstract

The distributional support of the sample paths of L\'evy processes is an important issue for the construction of sparse statistical models, theories of integration in infinite dimensions and the existence of generalized solutions of stochastic partial differential equations driven by L\'evy white noise. Here one considers a family K_\alpha (0<\alpha<2) of L\'evy processes which have no support in S'. For 1<\alpha<2 they are supported in K', the space of distributions of exponential type and for 0<\alpha=<1 on similar spaces of power exponential type.