In this paper, some new forms of Cheeger's inequalities are established for general (maybe unbounded) symmetric forms (Theorems 1.1 and 1.2), the resulting estimates improve and extend the ones obtained by Lawler and Sokal for bounded jump processes. Furthermore, some existence criteria for spectral gap of general symmetric forms or general reversible Markov processes are presented (Theorems 1.4 and 3.1), based on Cheeger's inequalities and a relationship between the spectral gap and the first Dirichlet and Neumann eigenvalues on local region.

This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some su

This note studies the first non-trivial eigenvalue of second order self-adjoint elliptic operators in Rd. Some lower bounds of the eigenvalue are obtained by using a probabilistic approach and some geometric consideration. In one-dimensional case, an analytic proof is also presented. The resulting bounds can be sharp.

This paper studies the equivalence of exponential ergodicity and L2-exponential convergence mainly for continuous-time Markov chains. In the reversible case, we show that the known criteria for exponential ergodicity are also criteria for L2-exponential convergence. Until now, no criterion for L2-exponential convergence has appeared in the literature. Some estimates for the rate of convergence of exponentially ergodic Markov chains are presented. These estimates are practical once the stationary distribution is known. Finally, the reversible part of the main result is extended to the Markov processes with general state space.

A general formula for the lower bound of the first eigenvalue on compact Riemannian mani-folds is presented in this paper for the first time. The formula improves the main known sharp estimates including Lichnerowicz's estimate and Zhong-Yang's estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on a probabilistic approach (i.e., the coupling method) introduced by the authors previously.