Abstract
The fractional Laplacian \((-\triangle)^{\gamma/2}\) 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 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 \((-\triangle)^{\gamma/2}\) 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 I γ f is not p-integrable. We then introduce the new unique left-inverse I γ, p of the fractional Laplacian \((-\triangle)^{\gamma/2}\) with the property that I γ, p is dilation-invariant (but not translation-invariant) and that I γ, p f is p-integrable for any Schwartz function f. We finally apply that linear operator I γ, 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.
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Communicated by Charles Micchelli.
Research of first author was supported by the National Science Foundation under grant DMS-1109063. Research of second author was supported by the Swiss National Science Foundation under grant 200020-121763.
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Sun, Q., Unser, M. Left-inverses of fractional Laplacian and sparse stochastic processes. Adv Comput Math 36, 399–441 (2012). https://doi.org/10.1007/s10444-011-9183-6
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DOI: https://doi.org/10.1007/s10444-011-9183-6
Keywords
- Fractional Laplacian
- Riesz potential
- Impulsive Poisson noise
- Fractional stochastic process
- Stochastic partial differential operator