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

Chen, Zhen-Qing; Kumagai, Takashi

doi:10.1016/s0304-4149(03)00105-4

Cited by 386 publications

(424 citation statements)

References 19 publications

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“…There are certain values of β for which the Dirichlet forms are known to be regular [7,16,17,42], and there are some standard examples where the values of β * can be determined explicitly [22,30]. These issues will be discussed in Section 7, and more detail in the forthcoming paper [29].…”

Section: For a Homogenous Ifs {Smentioning

confidence: 99%

“…For 0 < β < 2 , i.e., λ ∈ (r 2 , 1), it has been shown in [7] that the heat kernel satisfies the following estimate: p(t, ξ, η) ≍ min t −α/β , t |ξ − η| α+β , ξ, η ∈ K, 0 < t ≤ 1.…”

Section: Induced Dirichlet Formsmentioning

confidence: 99%

“…If the domain is dense in L 2 (M, ν), then E M defines a non-local Dirichlet form. In the analysis of fractals, there is a large literature on the study of local and non-local Dirichlet forms as well as their associated heat kernels on self-similar sets, d-sets and more general metric measure spaces [6,7,[16][17][18]22,26,27,38,39,42,43]. In many cases, a non-local form can be obtained by subordination of a local form, and has a jump kernel with order |ξ − η| −(α+β) , where α is the Hausdorff dimension of the underlying set and β is the walk dimension of the corresponding stable-like process [7,42].…”

Section: Introductionmentioning

confidence: 99%

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Random walks and induced Dirichlet forms on self-similar sets

Kong

Lau

Wong

2017

Advances in Mathematics

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Let K be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea [25], it has been proved that on the symbolic space X of K a natural augmented tree structure E exists; it is hyperbolic, and the hyperbolic boundary ∂ H X with the Gromov metric is Hölder equivalent to K. In this paper we consider certain reversible random walks with return ratio 0 < λ < 1 on (X, E). We show that the Martin boundary M can be identified with ∂ H X and K. With this setup and a device of Silverstein [41], we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ(ξ, η) ≍ |ξ − η| −(α+β) , where α is the Hausdorff dimension of K and β depends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form on K. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).

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Section: For a Homogenous Ifs {Smentioning

confidence: 99%

Section: Induced Dirichlet Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Random walks and induced Dirichlet forms on self-similar sets

Kong

Lau

Wong

2017

Advances in Mathematics

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“…One of the key steps in proving the upper and lower bounds in [4] is the parabolic Harnack inequality. Chen and Kumagai [9] showed that the parabolic Harnack inequality holds for symmetric stable-like processes in d sets and established upper and lower bounds on the transition densities of these processes. All the processes mentioned above satisfy a certain scaling property which was used crucially in the proofs of the Harnack inequalities.…”

Section: Introductionmentioning

confidence: 99%

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

2006

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Abstract. Let X be a Lévy process in R d , dU3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.Mathematics Subject Classifications (2000): Primary 60J45, 60J75, Secondary 60J25.

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“…One of the key steps in proving the upper and lower bounds in [3] is the parabolic Harnack inequality. In [5], Chen and Kumagai showed that the parabolic Harnack inequality holds for symmetric stable-like processes in d-sets and established upper and lower bounds on the transition densities of these processes. All the processes mentioned above satisfy a certain scaling property which was used crucially in the proofs of the Harnack inequalities.…”

mentioning

confidence: 99%