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In the present paper, we consider long time behaviors of the volume of theWiener sausage on Dirichlet spaces. We focus on the volume of the Wienersausage for diffusion processes on metric measure spaces other than the Euclidspace equipped with the Lebesgue measure. We obtain the growth rate of theexpectations and almost sure behaviors of the volumes of the Wiener sausages onmetric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussianheat kernel estimates. We show that the growth rate of the expectations on a"bounded" modification of the Euclidian space is identical with the one on theEuclidian space equipped with the Lebesgue measure. We give an example of ametric measure Dirichlet space on which a scaled of the means fluctuates.

Long time behavior of the volume of the Wiener sausage on Dirichlet spaces