Eigenvalues, inequalities and ergodic theory

By Mu-Fa Chen. 2001

224 citations recorded by the source corpus.

This paper surveys the main results obtained during the period 1992—1999 on three aspects mentioned in the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e. the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (§ 1). Here, a probabilistic method — coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger’s method which comes from Riemannian geometry. This consists of § 2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented (§ 3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a12f1cc5f9f662c9d07fda5d0a03d96d41c0d5f0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Approximation of Invariant Measures for Regime-switching Diffusions (2014, background): “, Chen [8]), existence and uniqueness of numerical invariant measures for regime-switching diffusions with multiplicative noises can also be addressed if the principal eigenvalue is positive.”
Bilateral Hardy-type inequalities (2012, background): “1 at hand, it is ready to write down some estimates of the optimal constant A in (1), as we did in [3, 5], in terms of B θ given below (cf.”
Extremal values of smallest eigenvalues of Hill's operators with potentials in L 1 balls ✩ (2009, background): “For example, the maximum principle has a close connection with positiveness of the flrst Dirichlet eigenvalues [2, 5]. Some nice works on estimates of eigenvalues are [1, 6, 7 , 8, 12, 14, 26, 28].”
Transportation-information inequalities for Markov processes (2007, background): “(2) The quantity C( ) in (4.16) is not innocent: Chen-Wang’s variational formula for the spectral gap tells us that ([ 11 , 41]): cP(µ) = inf C( ).”
FUNCTIONAL INEQUALITIES, SEMIGROUP PROPERTIES AND SPECTRUM ESTIMATES (2000, background): “There are also many books and survey papers in this literature, for instances [4, 10, 13, 15, 16, 19 , 20, 29, 31, 36] (see also the references therein).”
Poincaré inequalities and hitting times (2010, background): “Hence (3) holds for Q 2 , and this implies that the Poincaré inequality (4) holds according to an argument due to Mu-Fa Chen ([18], pp. 221–235).”
Large deviations of the empirical flow for continuous time Markov chains (2012, background): “that condition (9.2) is equivalent to recurrence (combine [36][Ex. 1.3.4] with [36][Th. 3.4.1]). Under the above assumptions, the logarithmic Sobolev inequality holds if and only if (see Table 1.4 in [12][Ch. 1]) sup k≥1 π[k,+∞) log 1 π[k,+∞) kX−1 j=0 1 π(j)bj <+∞. (9.4) Possible absence of expo”