No abstract

We prove the compactness of the imbedding of the Sobolev space W 1,2 0 (Ω) into L 2 (Ω) for any relatively compact open subset Ω of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Hölder continuous heat kernel. Subject Classification (1991): 53C70, 58J35, 58J50, 31C15, 31C25, 35K05, 53C20, 53C23 Mathematics

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained.Comment: Published in at http://dx.doi.org/10.1214/07-AOP347 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org); Errata DOI: 10.1214/11-AOP68

We construct the (1; p)-Sobolev spaces and energy functionals over L p-maps between metric spaces for p 1 under the condition so-called strong measure contraction property of Bishop-Gromov type. Under this property, we also prove the existence of energy measures, and the weak Poincar e inequality, which extends some parts of the results in Korevaar-Schoen and Sturm.

We establish the coincidence of two classes of Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of Kato class measures is defined by way of the heat kernel, another is defined in terms of the Green kernel depending on some exponents related to the heat kernel estimates. We also prove that pth integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of Kato class if p is greater than a constant related to the estimate under the same conditions. These are complete extensions of some results for the Brownian motion on Euclidean space by Aizenman and Simon. Our result can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.