A white noise approach to stochastic partial differential equations driven by the fractional L\xe9vy noise

Xuebin Lü,Wanyang Dai

doi:10.1186/s13662-018-1861-y

Xuebin Lü, Wanyang Dai

Open Access

https://doi.org/10.1186/s13662-018-1861-y

Copy DOI

Abstract

In this paper, based on the white noise theory for d-parameter Lévy random fields given by (Holden et al. in Stochastic Partial Differential Equations: A modeling, white noise functional approach, 2010), we develop a white noise frame for anisotropic fractional Lévy random fields to solve the stochastic Poisson equation and the stochastic Schrödinger equation driven by the d-parameter fractional Lévy noise. The solutions for the two kinds of equations are all strong solutions given explicitly in the Lévy–Hida stochastic distribution space.

Highlights

1 Introduction In recent years, fractional Lévy processes are getting popular since they are more flexible in modeling the distributions of noises than fractional Brownian motions (FBMs)
More and more researchers have been attracted to the studies of fractional Lévy processes, stochastic calculus for fractional Lévy processes, and stochastic differential equations driven by these processes
In [11], a white noise theory for fractional Lévy process was developed by considering it as a generalized functional of the sample path of Lévy process and solving several kinds of stochastic ordinary differential equations driven by fractional Lévy noises

Summary

Introduction

Fractional Lévy processes are getting popular since they are more flexible in modeling the distributions of noises than fractional Brownian motions (FBMs) They can capture large jumps and model high variability in the real systems appearing in finance, telecommunications, and so on. In [11], a white noise theory for fractional Lévy process was developed by considering it as a generalized functional of the sample path of Lévy process and solving several kinds of stochastic ordinary differential equations driven by fractional Lévy noises. The object of this paper is developing white noise theory for fractional Lévy random fields and the study of stochastic partial differential equations (SPDEs) driven by a fractional Lévy noise, that is, solving stochastic Poisson and Schrödinger equations driven.

Note that here we interpret all products as

Lévy random field is defined by

We denote by

Proof Since

Laplace operator given bounded domain in and

It follows