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.

Abstract. The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case and certain extension is made in the paper.

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

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.

Some estimates of logarithmic Sobolev constant for general symmetric forms are obtained in terms of new Cheeger's constants. The estimates can be sharp in some sense.