A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and maximal inequalities

A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and... 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 $$L_{2}$$ L 2 mixing time, $$\tau _{2}$$ τ 2 (while there are sophisticated analytic tools to bound $$\tau _2$$ τ 2 , in general they do not determine $$\tau _2$$ τ 2 up to a constant factor and they lack a probabilistic interpretation). In this work we show that $$\tau _2$$ 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, $$c_{{\mathrm {LS}}}$$ c LS , as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $$c_{{\mathrm {LS}}}$$ c LS in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $$\tau _2$$ τ 2 is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $$\tau _2$$ τ 2 is robust under bounded perturbations of the edge weights. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and maximal inequalities

, Volume 170 (4) – Mar 14, 2017
32 pages

/lp/springer_journal/a-characterization-of-l-2-l-2-mixing-and-hypercontractivity-via-O0ArbcVy1R
Publisher
Springer Journals
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0769-x
Publisher site
See Article on Publisher Site

Abstract

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 $$L_{2}$$ L 2 mixing time, $$\tau _{2}$$ τ 2 (while there are sophisticated analytic tools to bound $$\tau _2$$ τ 2 , in general they do not determine $$\tau _2$$ τ 2 up to a constant factor and they lack a probabilistic interpretation). In this work we show that $$\tau _2$$ 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, $$c_{{\mathrm {LS}}}$$ c LS , as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $$c_{{\mathrm {LS}}}$$ c LS in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $$\tau _2$$ τ 2 is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $$\tau _2$$ τ 2 is robust under bounded perturbations of the edge weights.

Journal

Probability Theory and Related FieldsSpringer Journals

Published: Mar 14, 2017

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations