Heat kernel estimates for stable-like processes on d-sets

By Zhen-Qing Chen, Takashi Kumagai. 2003

460 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53996ed097127755dda0bf28e2fc054b4518e804

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Heat Kernels on Metric Measure Spaces (2014, background): “1 Upper bounds for non-local Dirichlet forms The techniques for obtaining heat kernel bounds for non-local Dirichlet forms has been developed by a number of authors, see for example [7, 10, 14, 17, 18] and the references therein.”
Regularity of Harmonic functions for a class of singular stable-like processes (2009, unknown): “Associated with the operator L is the symbol ℓ(x, u) := c2 d∑ j=1 |u · aj(x)|α, x, u ∈ Rd.”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “1) holds for all t > 0 and all x, y ∈ X, see [33], and therefore we can take R0 =∞.”
On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces (2009, methodology): “13 in [4]) For any bounded Borel measurable function q(t, x) that is parabolic in an open subset D of R+×F , s 7→ q(Zs∧τD) = q(Zs∧τD(Y )) is right continuous P (t,x)-a.”
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps (2007, methodology): “BB]. The case CK , CK2].”
Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part (2009, background): “Here, unlike in [8], there is no restriction on β − α. • In a recent preprint [18], Chen, Kim and Kumagai looked at truncated jump processes whose kernel is given by the following J(x, y) = c(x, y) |x− y|d+α 1(|x−y|≤κ), where α ∈ (0, 2), κ is a positive constant and c(x, y) is bounded below and abov”
Harnack inequality for stable processes on d-sets (2003, background): “Kumagai [CK] studied general stable-like processes on fractals defined by means of appropriate Dirichlet forms.”
Regularity results for nonlocal parabolic equations (2013, background): “It is shown in [21] that the heat kernel p(t, x, y) exists and satisfies”
Approximation of stable law in Wasserstein-1 distance by Stein’s method (2017, background): “See [17] for more details about heat kernel estimates of stable type processes.”
Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators (2007, background): “Lemma 3 below is an example of such an estimate, see also [ 11 ].”
Transition Density Estimates for a Class of Lévy and Lévy-Type Processes (2009, background): “Pure jump processes whose jump kernel is comparable to the one of truncated stable-like process are considered by Chen, Kim and Kumagai [10].”
Regularity estimates for elliptic nonlocal operators (2015, unknown): “y k(x,·) which satisﬁes some isotropic lower bound, e.g. for some c0 > 0, α ∈ (0,2) µ(x,dy) = k(x,y)dy, k(x,y) ≥ c0|x−y|−d−α (|x−y| ≤ 1), then Theorem 1.2 is proved in resp. follows from the works [24, 4, 9, 8]. In these works the constant c in (1.14) depends on α ∈ (0,2) with c(α) → +∞ for α → 2”
Heat kernel estimates for subordinate Brownian motions (2015, methodology): “For pure jump symmetric processes with stable-like jumping kernels on Z or R sharp heat kernel estimates were obtained in [2, 8, 10], while in [6, 15, 16, 19, 28] they were obtained for some classes of Lévy processes.”
A new formulation of Harnack?s inequality for nonlocal operators (2011, unknown): “A priori estimates in Hölder spaces for similar Dirichlet forms have been obtained in [8,2,4] and other works.”