Fractional Laplacians on the sphere, the
                    Minakshisundaram zeta function and
                    semigroups

Nápoli, Pablo L. De; Stinga, P. R.

doi:10.1090/conm/723/14545

Cited by 6 publications

(11 citation statements)

References 19 publications

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“…This is an immediate consequence of the third formula for U in (20) and the results of Theorem 7, see [90,91]. For such explicit statement for negative powers L −s in other contexts like manifolds and discrete settings, see [34,39,45,70].…”

Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 90%

“…Consider the situation where we have derived a model (usually a nonlinear PDE problem) that involves a fractional power of some partial differential operator L. As we saw before, L can be a Laplacian or a heat operator, or even an operator on a manifold [7,39] or a lattice in the case of discrete models [34]. Then we are faced at least with the following basic questions.…”

Section: The Methods Of Semigroupsmentioning

confidence: 99%

“…It turns out these are quite concrete and useful ways of defining and understanding fractional operators. Indeed, when a heat kernel is available for the semigroup e −tL , then pointwise formulas for both positive and negative powers of L can be obtained, see [13,14,29,32,33,34,38,39,45,70,76,90,91,92,93,94]. For degenerate cases like the usual derivative or discrete derivatives see [1,11].…”

Section: The Methods Of Semigroupsmentioning

confidence: 99%

“…This Bessel function analysis was applied in the extension problem for other fractional operators in [11,14,29,39,73,90].…”

Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 99%

See 3 more Smart Citations

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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The method of semigroups is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators L and their regularity properties in related functional spaces. The approach was introduced by the author and José L. Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian L s = (−∆) s , 0 < s < 1. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for (−∆) s u(x). One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on u, the validity of the pointwise limitsThe formula for the solution to the Poisson problem (−∆) s u = f is found through the semigroup approach as the inverse of the fractional Laplacian u(x) = (−∆) −s f (x) (fundamental solution). We then present the Caffarelli-Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Hölder and Schauder estimates (−∆) ±s : C α → C α∓2s are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Hölder and Zygmund spaces with heat semigroups.

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Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 90%

Section: The Methods Of Semigroupsmentioning

confidence: 99%

Section: The Methods Of Semigroupsmentioning

confidence: 99%

“…This Bessel function analysis was applied in the extension problem for other fractional operators in [11,14,29,39,73,90].…”

Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 99%

See 2 more Smart Citations

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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“…See [64]. ) In the case of the unit circle in 2D plane, one can refer to [65]. For spheres in higher dimensions, s = 0.…”

Section: Charged Particles On the Spherementioning

confidence: 99%

Random Batch Methods (RBM) for interacting particle systems

Jin

Liu

2020

Journal of Computational Physics

115

189

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We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N 2 ) per time step to O(N ), for a system with N particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing N -independent time steps and also capture, in the N → ∞ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.

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Pointwise monotonicity of heat kernels

et al. 2021

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In this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.

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Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

Cited by 6 publications

References 19 publications

User’s guide to the fractional Laplacian and the method of semigroups

User’s guide to the fractional Laplacian and the method of semigroups

Random Batch Methods (RBM) for interacting particle systems

Pointwise monotonicity of heat kernels

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