There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the $L_{2}$ mixing time, $τ_{2}$ (while there are sophisticated analytic tools to bound $ τ_2$, in general they do not determine $τ_2$ up to a constant factor and they lack a probabilistic interpretation). In this work we show that $τ_2$ can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, $c_{\mathrm{LS}}$, as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $c_{\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $τ_2$ is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $τ_2 $ is robust under bounded perturbations of the edge weights.
Gradual smoothing: strong hypercontractivity and logarithmic Sobolev inequalities
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM)（数学科学研究所 ICMAT）；University of Technology Sydney (UTS)（悉尼科技大学）；Universidad Autónoma de Madrid（马德里自治大学）
In this work, we study the validity of Entropy inequalities under decorrelation assumptions at large-scales, and also we study the problem of Weighted Meyers estimate in perforated domains. As a consequence, we obtain the hypercontractivity of correctors with Neumann boundary conditions.
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on $S_n$, which are functions whose $2$-norm…
Poincare inequality and exponential integrability of the hitting times of a Markov process
推广了文献[Mathieu，P.（1997）Hitting times and spectral gap inequalities，Ann. Inst. Henri Poincare 33，4，437 - 465]的方法，证明了（可能非对称的）马尔可夫过程的Poincare不等式给出了该过程的首中时的指数可积性.对于对称椭圆扩散，这提供了Poincare不等式的首中时判据。
We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santaló inequality and the inverse Santaló inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's…
We present results relating mixing times to the intersection time of branching random walk (BRW) in which the logarithm of the expected number of particles grows at rate of the spectral-gap $\mathrm{gap}$ . This is a finite state space analog of a critical branching process. Namely, we show that the maximal expected hitting time of a state by such a BRW is up to a universal constant larger than the $L_{\infty}$ mixing-time, whereas under transitivity the same is true for the intersection time of two independent such BRWs. Using the same methodology, we show that for a sequence of reversible Ma…
