Heat kernel estimates for time fractional equations

AbstractIn this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

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On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

Acta Mathematica Sinica English Series ◽

10.1007/s10114-009-8576-7 ◽

2009 ◽

Vol 25 (7) ◽

pp. 1067-1086 ◽

Cited By ~ 21

Author(s):

Zhen-Qing Chen ◽

Panki Kim ◽

Takashi Kumagai

Keyword(s):

Heat Kernel ◽

Harnack Inequality ◽

Jump Processes ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Heat Kernel Estimates ◽

Parabolic Harnack Inequality ◽

Measure Spaces

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Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Memoirs of the American Mathematical Society ◽

10.1090/memo/1330 ◽

2021 ◽

Vol 271 (1330) ◽

Author(s):

Zhen-Qing Chen ◽

Takashi Kumagai ◽

Jian Wang

Keyword(s):

Heat Kernel ◽

Jump Processes ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Open Problems ◽

Content Type ◽

Heat Kernel Estimates ◽

Measure Spaces ◽

Non Local ◽

Volume Doubling

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α \alpha -stable-like processes even with α ≥ 2 \alpha \ge 2 when the underlying spaces have walk dimensions larger than 2 2 , which has been one of the major open problems in this area.

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

Probability Theory and Related Fields ◽

10.1007/s00440-007-0070-5 ◽

2007 ◽

Vol 140 (1-2) ◽

pp. 277-317 ◽

Cited By ~ 83

Author(s):

Zhen-Qing Chen ◽

Takashi Kumagai

Keyword(s):

Heat Kernel ◽

Jump Processes ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Heat Kernel Estimates ◽

Measure Spaces ◽

Mixed Types

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HARDY SPACES ON METRIC MEASURE SPACES WITH GENERALIZED SUB-GAUSSIAN HEAT KERNEL ESTIMATES

Journal of the Australian Mathematical Society ◽

10.1017/s144678871700012x ◽

2017 ◽

Vol 104 (2) ◽

pp. 162-194

Author(s):

LI CHEN

Keyword(s):

Hardy Space ◽

Heat Kernel ◽

Hardy Spaces ◽

Riesz Transform ◽

Square Function ◽

Space Theory ◽

Metric Measure Spaces ◽

Kernel Estimates ◽

Previous Theory ◽

Measure Spaces

Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviours (locally Gaussian and at infinity sub-Gaussian), in which case the previous theory does not apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the $H^{p}$ space corresponding to Gaussian estimates may not coincide with $L^{p}$. As a motivation for this theory, we show that the Riesz transform maps our Hardy space $H^{1}$ into $L^{1}$.

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