Heat Kernel Estimates On Ultra-metric Spaces

In this thesis,we focus on heat kernel estimates on ultra-metric spaces,and obatin heat kernel upper estimates of pure jump Dirichlet form by using the Davies method and heat kernel estimates of non-local operators with potentials by using the Feynman-Kac transform.This thesis is split into two parts.In the first part,we use the Davies method to give a quick proof for the heat kernel upper bound for the non-local Dirichlet form on ultra-metric spaces.Firstly,we use the on-diagonal upper estimates and tail estimates of jump kernel to obtain heat kernel upper estimates in homogeneous space.The key observation is that heat kernel estimates of truncated Dirichlet form vanishes when two spatial points are separated by any ball of radius larger than the truncated range,and this new phenomenon arises from the ultra-metric property of the space.Secondly,we use a new method to obtain heat kernel upper estimates with a general time-space scaling.Finally,we discuss some equivalent conditions of heat kernel upper estimates.In the second part,we consider Dirichlet heat kernel estimates of non-local operators with potentials on ultra-metric measure spaces.Firstly,we introduce n-dimensional padic number fields and construct the Markov process using ultra-metric property.Meanwhile we get the new equation about p-adic Schr ¨odinger-type operator.Secondly,from heat kernel estimates of non-local operator,we use the Feynman-Kac transform to obtain explicit estimates of Dirichlet heat kernel of non-local operators with potentials,which may not be in the appropriate Kato class.The key result is that we find a new example about heat kernel estimates of the Feynman-Kac semigroup.