Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes

Q. Sun, M. Unser

Advances in Computational Mathematics, vol. 36, no. 3, pp. 399-441, April 2012.

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The fractional Laplacian (−∆)^γ⁄2 commutes with the primary coordination transformations in the Euclidean space ℝ^d: 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 I_γ 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_γ 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 ƒ such that I_γƒ is not p-integrable. We then introduce the new unique left-inverse I_γ,p of the fractional Laplacian (−∆)^γ⁄2 with the property that I_γ,p is dilation-invariant (but not translation-invariant) and that I_γ,pƒ is p-integrable for any Schwartz function ƒ. We finally apply that linear operator I_γ,p with p = 1 to solve the stochastic partial differential equation (−∆)^γ⁄2Φ = w with white Poisson noise as its driving term w.

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AUTHOR="Sun, Q. and Unser, M.",
TITLE="Left-Inverses of Fractional {L}aplacian and Sparse Stochastic
	Processes",
JOURNAL="Advances in Computational Mathematics",
YEAR="2012",
volume="36",
number="3",
pages="399--441",
month="April",
note="")

DOI: 10.1007/s10444-011-9183-6