# Modelling Levy space-time white noises

## TL;DR

This paper extends Gaussian space-time white noise to Levy-type noise using Levy-valued random measures, characterizing their properties and embedding them into distribution spaces, thus broadening the mathematical framework for stochastic processes.

## Contribution

It introduces Levy-valued random measures as a generalization of white noise, characterizes their associated cylindrical Levy processes, and establishes their embedding in distribution spaces.

## Key findings

- Characterization of Levy-valued random measures via characteristic functions
- Embedding conditions into distribution spaces based on integrability
- Representation of Levy-valued measures as derivatives of Levy sheets

## Abstract

Based on the theory of independently scattered random measures, we introduce a natural generalisation of Gaussian space-time white noise to a Levy-type setting, which we call Levy-valued random measures. We determine the subclass of cylindrical Levy processes which correspond to Levy-valued random measures, and describe the elements of this subclass uniquely by their characteristic function. We embed the Levy-valued random measure, or the corresponding cylindrical Levy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. Similar to existing definitions, we introduce Levy-valued additive sheets, and show that integrating a Levy-valued random measure in space defines a Levy-valued additive sheet. This relation is manifested by the result, that a Levy-valued random measure can be viewed as the weak derivative of a Levy-valued additive sheet in the space of distributions.