For the measurable Riemannian structure on the Sierpinski gasket introduced by Kigami, various short time asymptotics of the associated heat kernel are established, including Varadhan's asymptotic relation, some sharp one-dimensional asymptotics at vertices, and a non-integer-dimensional on-diagonal behavior at almost every point. Moreover, it is also proved that the asymptotic order of the eigenvalues of the corresponding Laplacian is given by the Hausdorff and boxcounting dimensions of the space.

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a 'geometric counting function' defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.

ABSTRACT. The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure Mγ , formally written asGaussian free field X. It is an Mγ -symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure Mγ . In this paper we provide a detailed analysis of the heat kernel pt(x, y) of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form pt(x, y) ≤ C1t2 , and an on-diagonal lower bound of the] heavily dependent on x, for each η > 18 for Mγ -almost every x. As applications, we deduce that the pointwise spectral dimension equals 2 Mγ -a.e. and that the global spectral dimension is also 2.

This expository article is devoted to a survey of existent results concerning the measurable Riemannian structure on the Sierpiński gasket and to a brief account of the author's recent result on Weyl's eigenvalue asymptotics of its associated Laplacian. In particular, properties of the Hausdorff measure with respect to the canonical geodesic metric are described in some detail as a key step to the proof of Weyl's asymptotics. A complete characterization of minimal geodesics is newly proved and applied to invalidity of Ricci curvature lower bound conditions such as the curvature-dimension condition and the measure contraction property. Possibility of and difficulties in extending the results to other self-similar fractals are also discussed.

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