Jun Kigami - Kyoto University We are going to define a class of measures on the 2-dim. square called weakly exponential measures, which includes Liouville measures and statistically random measures. Then we will study time change of the Brownian motion with respect to this class of measures. Our aim is to establish Poincare inequality, Nash type inequality, ultracontractivity of semigroups, existence of continuous heat kernels and heat kernel estimates.