Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

The fractional Laplacian (−�)γ/2 commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential Iγ which is dilationinvariant 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 Iγ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (−�)γ/2 which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1− 1/p), there exists a Schwartz function f such that Iγf is not p-integrable. We then introduce the new unique leftinverse Iγ,p of the fractional Laplacian (−�)γ/2 with the property that Iγ,p is dilation-invariant (but not translation-invariant) and that Iγ,pf is p-integrable for any Schwartz function f . We finally apply that linear operator Iγ,p with p = 1 to solve the stochastic partial differential equation (−�)γ/2Φ = w with white Poisson noise as its driving term w.