Coupling, spectral gap and related topics (Ⅱ)  

作　　者：CHEN Muf aDepartment of Mathematics, Beijing Normal University, Beijing 100875, China 

出　　处：《Chinese Science Bulletin》1997年第17期1409-1416,共8页

摘　　要：THIS is the second one of a series of three reviews. The ideas introduced in the last review are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates is given, some new progress is reported and some open problems are proposed.THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the infinite-dimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).

关 键 词：DIFFUSIONS MANIFOLD MARKOV CHAINS spectral gap couplings. 

分 类 号：O211.6[理学—概率论与数理统计]