Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators

By Krzysztof Bogdan, Tomasz Jakubowski. 2007

249 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c312e8a3aab6dc1ae8a76d8e1a86bced5cff17e4

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise (2014, background): “In [5] this effect was rediscovered in a stronger form: it is required that b belongs to the Kato class Kd,α−1.”
Holder estimates for advection fractional-diffusion equations (2010, methodology): “In [2], heat kernel estimates are obtained using perturbation techniques for s ∈ (1/2, 1) for vector fields which belong just to a Kato class.”
Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, methodology): “This is due to the scaling, the rich range of asymptotic behaviors depending on the local geometry of the domain’s boundary, connections to the boundary Harnack principle and approximate factorization of the Green function, and applications in the perturbation theory of generators, in particular, vi”
Criteria for ergodicity of Lévy type operators in dimension one (2008, background): “For the Feller property and irreducibility about the symmetric stable process with drift term, refer to [7,23].”
Transition density estimates for jump Lévy processes (2010, background): “Jakubowski in [3] obtained estimates of heat kernels of the fractional Laplacian perturbed by gradient operators.”
Heat kernels for non-symmetric diffusion operators with jumps (2016, background): “In [3], B ogdan and Jakubowski considered a fundamental solution to the non-local operator ∆α/2+b(x)·∇with α ∈ (1, 2) andb belonging to some Kato’s class, and obtained its sharp two-sided estimat es.”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, methodology): “This explains the restriction 1 < α < 2 in [12], [34] and the present paper.”
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.”
Gradient estimate for Ornstein–Uhlenbeck jump processes (2010, background): “or the Lévy measure (see [8, 9, 12] and references within); see also [5, 11] for heat kernel upper bounds for α-stable processes with drifts.”
Stable process with singular drift (2014, background): “The fundamental solution p U n (t, x, y) of L n and p(t, x, y) are related by the following Duhamel's formula (see [8]): p U n (t, x, y) = p(t, x, y) +  t 0  R d p U n (s, x, z) U n (z) · ∇ z p(t − s, z, y)dzds.”
Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation (2010, background): “Throughout this paper we will assume that b = (b1, . . . , bd) is an Rd-valued function on Rd such that |b| ∈Kd,α−1.”
Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “It seems that our estimates would allow one to extend results from [4] and [8] (obtained there for the fractional Laplacian) to more general generators of Lévy processes.”