Heat Kernel Estimates For Symmetric Jump Processes And Their Applications

The thesis,as a contination of the work by Chen-Kumagai-Wang,is focused on heat kernel estimates for symmetric jump processes and their applications.It contains the following three aspects:In Chapter 1,we consider symmetric stable-like jump processes of mixed-type on metric measure spaces,which are strongly recurrent.We establish new equivalent conditions for the associated two-sided heat kernel estimates in terms of Resistance forms.The new equivalent conditions use two-sided estimates of both jumping kernels and Resistance forms,instead of cut-off Sobolev inequalities.Characterizations for upper bounds of heat kernel estimates as well as near diagonal lower bounds of Dirichlet heat kernel estimates are also established via Resistance forms.In Chapter 2,we aim to establish upper bounds of heat kernels for symmetric processes with finite range jumps on inhomogeneous metric measure spaces.We first apply super-Poincaré inequalities to establish local Nash-type inequalities as well as on-diagonal upper bounds of Dirichlet heat kernels.Then,by applying the properties of the exit time and the comparison inequality of Dirichlet heat kernels,we obtain upper bounds of global heat kernels from Dirichlet heat kernels.Our results indicate the precise dependencies between these estimates and the parameters involved in finite-range jumping kernels.In Chapter 3,we study two-sided estimates of transition probabilities for symmetric stable-like Markov chains of mixed-type on discrete spaces.By making full use of the relationship with the corresponding continuous time Markov chains as well as the parabolic Harnack inequalities,we establish stable equivalent characterizations of two-sided estimates of transition probabilities.In particular,this result is similar to those of Chen,Kumagai and Wang.