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.

By using a coupling technique, this paper presents some lower bounds of the first eigenvalue

One challenging problem in the context of reaction-diffusions is to prove the ergodicity or non-ergodicity for the Schl

This paper is devoted to studying a new topic: optimal Markovian couplings, mainly for time-continuous Markov processes. The study emphasizes the analysis of the coupling operators rather than the processes. Some constructions of optimal Markovian couplings for Markov chains and diffusions are presented, which are often unexpected. Then, the results are applied to study the L2-convergence for Markov chains and for a diffusion on compact manifold. The estimate of the convergent rate provided by this method can be sharp.

A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling[4], the Poincare inequality with respect to the absolute distribution of the process is also