Linear non-local diffusion problems in metric measure spaces

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

Download Full-text

Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

Journal of Functional Analysis ◽

10.1016/j.jfa.2017.01.001 ◽

2017 ◽

Vol 272 (8) ◽

pp. 3311-3346 ◽

Cited By ~ 6

Author(s):

Alexander Grigor'yan ◽

Eryan Hu ◽

Jiaxin Hu

Keyword(s):

Dirichlet Forms ◽

Heat Kernels ◽

Metric Measure Spaces ◽

Measure Spaces ◽

Non Local

Download Full-text

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.

Download Full-text

The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

Forum Mathematicum ◽

10.1515/forum-2017-0072 ◽

2018 ◽

Vol 30 (5) ◽

pp. 1129-1155 ◽

Cited By ~ 1

Author(s):

Jiaxin Hu ◽

Xuliang Li

Keyword(s):

Heat Kernel ◽

Upper Bound ◽

Dirichlet Form ◽

Measure Space ◽

Dirichlet Forms ◽

Upper Bounds ◽

Metric Measure Spaces ◽

Unified Approach ◽

Measure Spaces ◽

Non Local

AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

Download Full-text

Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

Mathematische Annalen ◽

10.1007/s00208-021-02350-z ◽

2022 ◽

Author(s):

Frederic Weber ◽

Rico Zacher

Keyword(s):

Heat Kernel ◽

General Framework ◽

Important Application ◽

Reduction Principle ◽

Non Local ◽

Beta 2 ◽

Local Diffusion ◽

Continuous Setting ◽

Discrete Setting ◽

Diffusion Problems

AbstractWe establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Download Full-text