The Conservativeness Of Dirichlet Forms On Metric Measure Spaces And Equivalent Conditions Of Upper Estimates Of Heat Kernels

This thesis studies the conservativeness of Dirichlet forms on metric measure spaces and some equivalent conditions of upper estimates of heat kernels.Firstly,we introduce some basic properties of the heat semigroup and the heat kernels.We then prove the conservetiveness from the survival estimates for strongly local Dirichlet forms.We further give a proof under a weaker condition for the conservativeness of Dirichlet forms without the killing terms on bounded or unbounded metric spaces.Finally,we give several equivalent conditions for upper estimates of heat kernels on bounded metric spaces.