Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

By Zhen-Qing Chen, Panki Kim, Renming Song. 2010

110 citations recorded by the source corpus.

Suppose that d≥2 and α∈(1,2). Let D be a bounded C1,1 open set in Rd and b an Rd-valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb=Δα/2+b⋅∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/d229d1cf8cc1bc4a47a2fa19cbe7e1915b41e35b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

Uniqueness of stable processes with drift (2013, methodology): “However in both [3] and [4], neither the uniqueness of fundamental solution qb(t, x, y) to Lb nor the uniqueness of the martingale problem for (Lb, C∞ c (Rd)) are addressed.”
Heat kernels for non-symmetric diffusion operators with jumps (2016, result): “The proof of the following result is similar to that of [10].”
Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “Song in [8] demand pointwise gradient estimates of transition densities and a Green function exactly of the type presented in our paper.”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, background): “In the case L = (∆|Ω) s + b · ∇ where b is a vector valued function belonging to a suitable Kato class, we refer to [20] for further details.”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, result): “Added in Proof The above mentioned results are available on arXiv [17].”
Stable process with singular drift (2014, background): “When α ∈ (1, 2) and the drifts are absolutely continuous with respect to the Lebesgue measure, sharp two-sided estimates on the heat kernels corresponding to the generators of the solutions to such SDEs were obtained in Rd [8,16,17] and in bounded C1,1 open sets [9].”