• Corpus ID: 17197138

From the long jump random walk to the fractional Laplacian

@inproceedings{Valdinoci2009FromTL,
  title={From the long jump random walk to the fractional Laplacian},
  author={Enrico Valdinoci},
  year={2009}
}
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. 
View PDF on arXiv
257 Citations
Highly Influential Citations
8
Background Citations
135
Methods Citations
23
Results Citations
3

257 Citations

Continuous random walks and fractional powers of operators

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes.

Competing power problem involving the half Laplacian

We prove the existence and the limit profile of the least energy solution of a half Laplacian equation with competing powers.

On a fractional reaction-diffusion models arising in population dynamics

This paper is concerned with the existence of positive solutions for a fractional population model with the homogeneous Dirichlet condition on the exterior of a bounded domain. The approach is based…

Nodal bound state of nonlinear problems involving the fractional Laplacian

This paper is concerned with the following nonlinear fractional Schrödinger equation ε2s(−Δ)su+V(x)u=|u|p−2u,inRN,

A SIGN-CHANGING SOLUTION FOR NONLINEAR PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN

In this article, we establish the existence of a least energy signchanging solution for nonlinear problems involving the fractional Laplacian. Our main tool is constrained minimization in a closed…

The Dyadic Fractional Diffusion Kernel as a Central Limit

We obtain the fundamental solution kernel of dyadic diffusions in ℝ+ as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution…

The Dyadic Fractional Diffusion Kernel as a Central Limit

We obtain the fundamental solution kernel of dyadic diffusions in ℝ+ as a central limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution…

Existence and multiplicity results for Dirichlet problem with fractional Laplacian and nonlinearity

The existence and multiplicity results for Dirichlet BVPs with the fractional Laplacian are established depending on the range of parameter and behavior of the nonlinearity at zero and at infinity.

A fractional eigenvalue problem in $\mathbb{R}^N$

We prove that a linear fractional operator with an asymptotically constant lower order term in the whole space admits eigenvalues.

Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.
...

References

SHOWING 1-10 OF 51 REFERENCES

Fractional Brownian motion via fractional Laplacian

Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance

Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable…

The random walk's guide to anomalous diffusion: a fractional dynamics approach

The spectrum of positive elliptic operators and periodic bicharacteristics

Let X be a compact boundaryless C ∞ manifold and let P be a positive elliptic self-adjoint pseudodifferential operator of order m>0 on X. For technical reasons we will assume that P operates on…

Lectures on Stochastic Processes

3 4 CONTENTS 6 Markov chains: stationary distributions 35 6.

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

Abstract We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral…

Elliptic PDEs with Fibered Nonlinearities

In ℝm×ℝn−m, endowed with coordinates x=(x′,x″), we consider bounded solutions of the PDE $$\Delta u(x)=f(u(x))\chi(x').$$ We prove a geometric inequality, from which a symmetry result follows.

Stable non-Gaussian random processes

The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the…

Expected Number of Distinct Sites Visited by a Random Walk with an Infinite Variance

Consider a random walk of n steps on an infinite, simple cubic lattice. Let p(r) be the (symmetric) probability of a vector jump r, and let Sn be the expected number of distinct lattice points…

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

...