Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

TL;DR

This paper extends the classical Riesz potential to non-integer orders larger than the dimension, characterizes its invariance properties, and applies it to solve stochastic PDEs driven by Poisson noise.

Contribution

It introduces a new class of inverse operators for the fractional Laplacian with specific invariance and integrability properties, and applies these to stochastic PDEs.

Findings

Extended the Riesz potential to orders larger than the dimension.

Identified a unique left-inverse with dilation invariance and integrability.

Applied the inverse operator to solve stochastic PDEs with Poisson noise.

Abstract

The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary coordination transformations in the Euclidean space $\RR^d$: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For $0<\gamma<d$, its inverse is the classical Riesz potential $I_\gamma$ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential $I_\gamma$ to any non-integer number $\gamma$ larger than $d$ and show that it is the unique left-inverse of the fractional Laplacian $(-\triangle)^{\gamma/2}$ which is dilation-invariant and translation-invariant. We observe that, for any $1\le p\le \infty$ and $\gamma\ge d(1-1/p)$, there exists a Schwartz function $f$ such that $I_\gamma f$ is not $p$-integrable. We then introduce the new unique left-inverse $I_{\gamma, p}$ of the fractional Laplacian $(-\triangle)^{\gamma/2}$ with the property that $I_{\gamma, p}$ is dilation-invariant (but not translation-invariant) and that $I_{\gamma, p}f$ is $p$-integrable for any Schwartz function $f$. We finally apply that linear operator $I_{\gamma, p}$ with $p=1$ to solve the stochastic partial differential equation $(-\triangle)^{\gamma/2} \Phi=w$ with white Poisson noise as its driving term $w$.