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The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primarycoordination transformations in the Euclidean space $\RR^d$: dilation,translation and rotation, and has tight link to splines, fractals and stableLevy 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 invarianceproperties. In particular, we extend the definition of the classical Rieszpotential $I_\gamma$ to any non-integer number $\gamma$ larger than $d$ andshow that it is the unique left-inverse of the fractional Laplacian$(-\triangle)^{\gamma/2}$ which is dilation-invariant andtranslation-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 $I_\gammaf$ is not $p$-integrable. We then introduce the new unique left-inverse$I_{\gamma, p}$ of the fractional Laplacian $(-\triangle)^{\gamma/2}$ with theproperty that $I_{\gamma, p}$ is dilation-invariant (but nottranslation-invariant) and that $I_{\gamma, p}f$ is $p$-integrable for anySchwartz 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$w$.

Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes