Open Access

# Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$

### Abstract

In many models of Mathematical Physics, based on the study of a Markov chain $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ on ${\mathbb Z}^d$, one can prove by perturbative arguments a contraction property of the stochastic operator restricted to a subspace of local functions $\mathcal H_{M}$ endowed with a suitable norm. We show, on the example of a model of random walk in random environment with mutual interaction, that the condition is enough to prove a Central Limit Theorem for sequences $\{f(S^{k}\widehat \eta)\}_{k=0}^{\infty}$, where $S$ is the time shift and $f$ is strictly local in space and belongs to a class of functionals related to the H\"older continuous functions on the torus $T^{1}$.

Key words: Markov chains, stochastic operator, mixing, random walks in random environment.

### Article Information

 Title Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$ Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 302-314 MathSciNet MR3469530 zbMATH 06630276 Milestones Received 17/04/2015 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

C. Boldrighini
Dipartimento di Matematica G. Castelnuovo, Sapienza Universita di Roma, Piazzale Aldo Moro 5, 00185 Roma; GNFM, Istituto Nazionale di Alta Matematica, Piazzale Aldo Moro 5, 00185 Roma

A. Marchesiello
Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Decin Branch, Pohranicni 1, 40501 Decin

C. Saffirio
Institut fur Mathematik Universit at Zurich, Winterthurerstrasse 190, CH-8057 Zurich

### Citation Example

C. Boldrighini, A. Marchesiello, and C. Saffirio, Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$, Methods Funct. Anal. Topology 21 (2015), no. 4, 302-314.

### BibTex

@article {MFAT789,
AUTHOR = {Boldrighini, C. and Marchesiello, A. and Saffirio, C.},
TITLE = {Weak dependence for a class of local functionals of Markov chains on ${\mathbb Z}^d$},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {4},
PAGES = {302-314},
ISSN = {1029-3531},
MRNUMBER = {MR3469530},
ZBLNUMBER = {06630276},
URL = {http://mfat.imath.kiev.ua/article/?id=789},
}

### References

1. Nicolae Angelescu, Robert A. Minlos, Valentin A. Zagrebnov, The one-particle energy spectrum of weakly coupled quantum rotators, J. Math. Phys. 41 (2000), no. 1, 1-23.  MathSciNet CrossRef
2. Carlo Boldrighini, Robert A. Minlos, Alessandro Pellegrinotti, Interacting random walk in a dynamical random environment. I. Decay of correlations, Ann. Inst. H. Poincar\e Probab. Statist. 30 (1994), no. 4, 519-558.  MathSciNet
3. Carlo Boldrighini, Robert A. Minlos, Alessandro Pellegrinotti, Interacting random walk in a dynamical random environment. II. Environment from the point of view of the particle, Ann. Inst. H. Poincar\e Probab. Statist. 30 (1994), no. 4, 559-605.  MathSciNet
4. C. Boldrighini, R. A. Minlos, A. Pellegrinotti, Random walk in a fluctuating random environment with Markov evolution, in: On Dobrushins way. From probability theory to statistical physics, Amer. Math. Soc., Providence, RI, 2000.  MathSciNet
5. K. Boldrigini, R. A. Minlos, A. Pellegrinotti, Random walks in a random (fluctuating) environment, Uspekhi Mat. Nauk 62 (2007), no. 4(376), 27-76.  MathSciNet CrossRef
6. K. Boldrigini, R. A. Minlos, A. Pellegrinotti, Random walks in random environment with Markov dependence on time, Condensed Matter Physics 11 (2008), no. 2, 209-221. CrossRef
7. Ju. A. Davydov, Mixing conditions for Markov chains, Teor. Verojatnost. i Primenen. 18 (1973), 321-338.  MathSciNet
8. Dmitry Dolgopyat, Gerhard Keller, Carlangelo Liverani, Random walk in Markovian environment, Ann. Probab. 36 (2008), no. 5, 1676-1710.  MathSciNet CrossRef
9. Dmitry Dolgopyat, Carlangelo Liverani, Non-perturbative approach to random walk in Markovian environment, Electron. Commun. Probab. 14 (2009), 245-251.  MathSciNet CrossRef
10. N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414.  MathSciNet
11. I. A. Ibragimov, Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971.  MathSciNet
12. C. Kipnis, S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104 (1986), no. 1, 1-19.  MathSciNet
13. Yu. G. Kondratiev, R. A. Minlos, One-particle subspaces in the stochastic $XY$ model, J. Statist. Phys. 87 (1997), no. 3-4, 613-642.  MathSciNet CrossRef
14. Yuri Kondratiev, Robert Minlos, Elena Zhizhina, One-particle subspace of the Glauber dynamics generator for continuous particle systems, Rev. Math. Phys. 16 (2004), no. 9, 1073-1114.  MathSciNet CrossRef
15. V. A. Malyshev, R. A. Minlos, Linear infinite-particle operators, American Mathematical Society, Providence, RI, 1995.  MathSciNet