Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow "intrinsic" with respect; to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, we consider the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. Our main concerns are following two problems: (I) When and how can we find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes? (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes? Note that in general stochastic processes associated with Dirichlet forms have jumps, i. e. paths of such processes are not continuous. The answer to (I) is for measures to have volume doubling property with respect to the resistance metric associated with a resistance form. Under volume doubling property, a new metric which is quasisymmetric with respect to the resistance metric is constructed and the Li-Yau type diagonal sub-Gaussian estimate of the heat kernel associated with the process using the new metric is shown. About the question (II), we will propose a condition called annulus comparable condition, (ACC) for short. This condition is shown to be equivalent to the existence of a good diagonal heat kernel estimate. As an application, asymptotic behaviors of the traces of 1-dimensional alpha-stable processes are obtained. In the course of discussion, considerable numbers of pages are spent on the theory of resistance forms and quasisymmetric maps.