Transition densities for Brownian motion on the Sierpinski carpet

Abstract

Upper and lower bounds are obtained for the transition densities p(t, x, y) of Brownian motion on the Sierpinski carpet. These are of the same form as those which hold for the Sierpinski gasket. In addition, the joint continuity of p(t, x, y) is proved, the existence of the spectral dimension is established, and the Einstein relation, connecting the spectral dimension, the Hausdorff dimension and the resistance exponent, is shown to hold. © 1992 Springer-Verlag.