Heat kernel estimates for the Dirichlet fractional Laplacian

By Zhen-Qing Chen, Panki Kim, Renming Song. 2009

264 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f148a7b80878913468a54a3b78e3bca7c936301d

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Heat kernel estimates for the fractional Laplacian with Dirichlet conditions (2009, methodology): “[10] based on the cases of C1,1 domains [19] and circular cones [10].”
Distributed-order fractional diffusions on bounded domains (2009, unknown): “[10] and Song and Vondrac̆ek [35].”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, background): “the linear fractional heat equation has a long tradition in Probability since fractional Laplacians are the infinitesimal generators of stable Lévy processes [1, 5, 28].”
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient (2010, background): “In Dresden Professor Renming Song announced related sharp estimates of heat kernel p̃D for bounded C1,1 open sets, analogous to [18].”
Regularity theory for general stable operators: parabolic equations (2015, background): “−1,0)× (Ω∩ B1) u = 0 in (−1,0)× B1 \Ω. (1.9) First, we prove a Cs x regularity estimate up to the boundary. For the fractional Laplacian this could be deduced combining the heat kernel estimates from [CKS10] with known interior estimates. However, such precise heat kernel estimates are not known for”
Dirichlet heat kernel estimates for rotationally symmetric Lévy processes (2013, unknown): “Unlike the cases considered in [4, 11, 13, 15], the characteristic exponents of the symmetric Lévy processes considered in this paper are quite general, satisfying only certain mild growth condition at ∞.”
Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians (2009, background): “It is proved in [ 12 ] that for a C 1,1 domain D, t ∈ (0, 1] and x, y ∈ D, we have”
Regularity results for nonlocal parabolic equations (2013, background): “In [19] sharp twosided estimates for the transition density (heat kernel) of the symmetric α–stable process (the fractional Laplacian) killed upon exiting a C1,1 open set Ω ⊂ Rd are obtained.”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “, [3, 10]; an explicit expression of the Poisson kernel for a ball [18]; and the explicit solution to problem (1.”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, background): “The fractional Laplacian operators of the form (−∆), σ ∈ (0, 2), are actually the infinitesimal generators of stable Lévy processes [6, 16, 44].”
Spectral properties of the Cauchy process on half‐line and interval (2009, methodology): “d CAUCHY PROCESS ON HALF-LINE AND INTERVAL 3 on exiting D= (0,∞) (or the heat kernel for − q − d2 dx2 with Dirichlet exterior condition on Dc), see Theorem 4 in Section 7. This extends the results of [8, 9, 13], where two-sided estimates for pD t (x,y) were obtained (in a more general setting). As a”
Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation (2010, methodology): “Clearly, many new and major difficulties arise when adapting the strategy outlined in [9] to Xb.”
Two-sided heat kernel estimates for censored stable-like processes (2010, unknown): “A (rotationally) symmetric α-stable process Y = {Y t , P x } on R d is a Lévy process such that E x e iξ ·(Y t −Y 0 ) = e −t|ξ | α for every x ∈ R d and ξ ∈ R”
Dirichlet heat kernel for unimodal Lévy processes (2014, methodology): “In a recent work [17], Chen, Kim and Song provide estimates of Dirichlet heat kernels for a class of pure-jump Markov processes with intensity of jumps comparable to that of a complete subordinate Brownian motion with scaling.”
Potential theory of subordinate Brownian motions revisited (2011, background): “r [27] and can be regarded as a preparation for [27] in this sense. Another omission is the Dirichlet heat kernel estimates of subordinate Brownian motions in smooth open sets recently established in [8, 9, 10, 11]. One of the reasons we do not include these recent results in this paper is that all ”
Symmetric jump processes and their heat kernel estimates (2009, methodology): “In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.”