The Domain of Definition of the L\'evy White Noise

TL;DR

This paper unifies two approaches to defining Le9vy white noise, extends its domain of definition, and provides practical criteria for its application in stochastic partial differential equations.

Contribution

It extends the definition of Le9vy white noise to an independently scattered random measure, unifying existing frameworks and enlarging their domain of applicability.

Findings

Unified the frameworks of Le9vy white noise as a generalized process and as a random measure.

Provided integrability conditions that enlarge the domain of definition for Le9vy white noises.

Established criteria for the existence of solutions to SPDEs driven by Le9vy white noise.

Abstract

It is possible to construct L\'evy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two approaches by extending the L\'evy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for L\'evy white noises, thereby maximally enlarging their domain of definition. Based on this connection, we provide new criteria for the practical determination of the domain of definition, including specific results for the subfamilies of Gaussian, symmetric-{\alpha}-stable, generalized Laplace, and compound Poisson white noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a L\'evy white noise.