From the long jump random walk to the fractional Laplacian

By Enrico Valdinoci. 2009

309 citations recorded by the source corpus.

This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/1b2f382e17ebe5b8da7bdf9fd652013475c062b2

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In Applied Mathematics

In Computational Theory and Mathematics

In Modeling and Simulation

Citations in the corpus

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations (2015, background): “In the theory of stochastic processes, this operator appears as the infinitesimal generator of a stable Lévy process [3, 45].”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “Un enfoque sencillo y muy bonito considerando caminos aleatorios con saltos largos puede verse en [86].”
Fractional diffusion with Neumann boundary conditions: the logistic equation (2012, methodology): “As the dynamic of a population dispersing via random walk is well described by a local operator –typically the Laplacian– Lévy diffusion processes are generated in the entire space by fractional powers of the Laplacian (−∆) for s ∈ (0, 1), as well explained in [26].”
Existence and stability of standing waves for nonlinear fractional Schr\"odinger equations with Hartree type nonlinearity (2012, background): “The Lévy processes, occuring widely in physics, chemistry and biology, lead to equations with the fractional Laplacians which have been recently studied by [1][8][23].”
Nonlocal problems with Neumann boundary conditions (2014, background): “Denoting u(x, t) the probability density of the position of the particle at time t > 0, with a similar discretization argument as in [21], one can see that u solves the fractional heat equation ut + (−∆) u = 0 in Ω for t > 0, with Nsu = 0 in R n \ Ω for t > 0.”
Some constructions for the fractional Laplacian on noncompact manifolds (2012, background): “In particular, nonlinear or free boundary problems involving fractional powers of the Laplacian (−Δ)γ appear naturally in applications (see for instance [67], [62] and the references therein).”
Hopf’s lemma and constrained radial symmetry for the fractional Laplacian (2016, background): “…ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, thin obstacle problem, optimal transport, image reconstruction, as well as diffusion problems (see, for instance, [2,9,28] and the references therein).”
BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL NONLOCAL FRACTIONAL LAPLACIAN PROBLEMS (2016, unknown): “, the papers [2, 7, 10, 16, 21, 34, 35, 36] and references therein).”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, background): “The interest in these operators has a long history in Probability since the fractional Laplacian operators of the form (−∆)σ/2, σ ∈ (0,2), are infinitesimal generators of stable Lévy processes [4, 14], see also [64].”
Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators (2013, background): “5) These kind of obstacle problems have been extensively studied in [3], [17] and [24] (see also [11], [26] and [27] for the basic definitions and properties of the fractional Laplacian).”
Regularity of solutions of the fractional porous medium flow with exponent 1/2 (2014, unknown): “[14, 17, 19].”