Title

Promoting University Startups: International Patterns, Vicarious Learning And Policy Implications

Keywords

Fractional Laplacian; Fractional stochastic process; Impulsive Poisson noise; Riesz potential; Stochastic partial differential operator

Abstract

The fractional Laplacian (-Δ) 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 (-Δ) 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 I γf is not p-integrable. We then introduce the new unique left-inverse I γ,p of the fractional Laplacian (-Δ) γ/2 with the property that I γ,p is dilation-invariant (but not translation-invariant) and that I γ, pf 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 (-Δ) γ/2 Φ = w with white Poisson noise as its driving term w. © 2011 The Author(s).

Publication Date

4-1-2012

Publication Title

Journal of Technology Transfer

Volume

37

Issue

3

Number of Pages

213-233

Document Type

Review

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s10961-010-9194-3

Socpus ID

84858450866 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84858450866

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