Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiaxin Hu ◽  
Guanhua Liu
Keyword(s):  
Dirichlet Form ◽  
Survival Function ◽  
Dirichlet Forms ◽  
Mean Value ◽  
Heat Kernels ◽  
Heat Semigroup ◽  
New Approach ◽  
Mean Value Inequality ◽  
Non Local ◽  
Upper Estimate

Abstract In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L 2 {L^{2}} -norm estimate of the survival function 1 - P t B ⁢ 1 B {1-P_{t}^{B}1_{B}} for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L 2 {L^{2}} mean-value inequality is borrowed to use.

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

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

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

2018 ◽  
Vol 30 (5) ◽  
pp. 1129-1155 ◽  
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.

scholarly journals Estimates of heat kernels for non-local regular Dirichlet forms

2014 ◽  
Vol 366 (12) ◽  
pp. 6397-6441 ◽  
Author(s):  
Alexander Grigor’yan ◽  
Jiaxin Hu ◽  
Ka-Sing Lau
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Non Local

scholarly journals NASH-TYPE INEQUALITIES AND HEAT KERNELS FOR NON-LOCAL DIRICHLET FORMS

2006 ◽  
Vol 60 (2) ◽  
pp. 245-265 ◽  
Author(s):  
Jiaxin HU ◽  
Takashi KUMAGAI
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Non Local

scholarly journals Heat kernels of the discrete Laguerre operators

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Aleksey Kostenko
Keyword(s):  
Hardy Inequality ◽  
Jacobi Polynomials ◽  
Heat Kernels ◽  
The Other ◽  
Heat Semigroup ◽  
Stochastic Completeness ◽  
Markovian Semigroup ◽  
Other Hand ◽  
The One

AbstractFor the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

A new simpler approach to measure the strength of toe plantar flexion requiring no mechanical restraint with a light-weight device

2021 ◽  
pp. 095441192110029
Author(s):  
Yuko Komuro ◽  
Yuji Ohta
Keyword(s):  
Plantar Flexion ◽  
Mean Value ◽  
Simple Approach ◽  
Light Weight ◽  
New Approach ◽  
Mean Values ◽  
Mechanical Restraint ◽  
Significant Difference ◽  
Ankle Joints ◽  
The Mean

Conventionally, the strength of toe plantar flexion (STPF) is measured in a seated position, in which not only the target toe joints but also the knee and particularly ankle joints, are usually restrained. We have developed an approach for the measurement of STPF which does not involve restraint and considers the interactions of adjacent joints of the lower extremities. This study aimed to evaluate this new approach and comparing with the seated approach. A thin, light-weight, rigid plate was attached to the sole of the foot in order to immobilize the toe area. Participants were 13 healthy young women (mean age: 24 ± 4 years). For measurement of STPF with the new approach, participants were instructed to stand, raise the device-wearing leg slightly, plantar flex the ankle, and push the sensor sheet with the toes to exert STPF. The sensor sheet of the F-scan II system was inserted between the foot sole and the plate. For measurement with the seated approach, participants were instructed to sit and push the sensor with the toes. They were required to maintain the hip, knee, and ankle joints at 90°. The mean values of maximum STPF of the 13 participants obtained with each approach were compared. There was no significant difference in mean value of maximum STPF when the two approaches were compared (new: 59 ± 23 N, seated: 47 ± 33 N). The coefficient of variation of maximum STPF was smaller for data obtained with the new approach (new: 39%, seated: 70%). Our simple approach enables measurement of STPF without the need for the restraints that are required for the conventional seated approach. These results suggest that the new approach is a valid method for measurement of STPF.

Basic Properties and Examples of Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Potential Theory ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Basic Properties ◽  
Time Changes ◽  
Symmetric Operators ◽  
Analytic Potential

This chapter introduces the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and presents their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. The chapter then presents some basic examples of Dirichlet forms, with special attention paid to their basic properties as well as explicit expressions of the corresponding extended Dirichlet spaces. Hereafter the chapter discusses the analytic potential theory for regular Dirichlet forms, and presents some conditions for the demonstrated Dirichlet form (E,F) to be local.

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