TY - JOUR

T1 - Heat kernel estimates for jump processes of mixed types on metric measure spaces

AU - Chen, Zhen Qing

AU - Kumagai, Takashi

N1 - Funding Information:
The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.

PY - 2008/1

Y1 - 2008/1

N2 - In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity e-c0 (x, y)|x-y|}, ∫α1α2} c(α, x, y)|x-y|d+α}, ν (dα) where ν is a probability measure on [α1, α2]subset (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ1 and γ2, where either γ2 gamma;1 > 0 or γ1 = γ2 = 0. This example contains mixed symmetric stable processes on Rn as well as mixed relativistic symmetric stable processes on Rn. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

AB - In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity e-c0 (x, y)|x-y|}, ∫α1α2} c(α, x, y)|x-y|d+α}, ν (dα) where ν is a probability measure on [α1, α2]subset (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ1 and γ2, where either γ2 gamma;1 > 0 or γ1 = γ2 = 0. This example contains mixed symmetric stable processes on Rn as well as mixed relativistic symmetric stable processes on Rn. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

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U2 - 10.1007/s00440-007-0070-5

DO - 10.1007/s00440-007-0070-5

M3 - Article

AN - SCOPUS:36148997271

SN - 0178-8051

VL - 140

SP - 277

EP - 317

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -