The Annals of Probability

Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices

Gady Kozma and Bálint Tóth

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a central limit theorem under diffusive scaling for the displacement of a random walk on $\mathbb{Z}^{d}$ in stationary and ergodic doubly stochastic random environment, under the ${\mathscr{H}_{-1}}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in $\mathscr{L}^{\max\{2+\delta,d\}}$, with $\delta>0$. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].

Article information

Ann. Probab., Volume 45, Number 6B (2017), 4307-4347.

Received: November 2014
Revised: October 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G99: None of the above, but in this section 60K37: Processes in random environments

Random walk in random environment central limit theorem Kipnis–Varadhan theory sector condition


Kozma, Gady; Tóth, Bálint. Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices. Ann. Probab. 45 (2017), no. 6B, 4307--4347. doi:10.1214/16-AOP1166.

Export citation


  • [1] Arveson, W. (1976). An Invitation to $C^{*}$-Algebras. Graduate Texts in Mathematics 39. Springer, New York.
  • [2] Bartholdi, L. and Erschler, A. Poisson-Fürstenberg boundary and growth of groups. Preprint. Available at arXiv:1107.5499.
  • [3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  • [4] Biskup, M. (2011). Recent progress on the random conductance model. Probab. Surv. 8 294–373.
  • [5] Carlen, E. (2010). Trace inequalities and quantum entropy: An introductory course. In Entropy and the Quantum (R. Sims and D. Ueltschi, eds.). Contemp. Math. 529 73–140. Amer. Math. Soc., Providence, RI.
  • [6] Chavel, I. (2001). Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics 145. Cambridge Univ. Press, Cambridge.
  • [7] Helland, I. S. (1982). Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9 79–94.
  • [8] Horváth, I., Tóth, B. and Vető, B. (2012). Relaxed sector condition. Bull. Inst. Math. Acad. Sin. (N.S.) 7 463–476.
  • [9] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [10] Komorowski, T., Landim, C. and Olla, S. (2012). Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 345. Springer, Heidelberg.
  • [11] Komorowski, T. and Olla, S. (2002). On the superdiffusive behavior of passive tracer with a Gaussian drift. J. Stat. Phys. 108 647–668.
  • [12] Komorowski, T. and Olla, S. (2003). A note on the central limit theorem for two-fold stochastic random walks in a random environment. Bull. Pol. Acad. Sci. Math. 51 217–232.
  • [13] Komorowski, T. and Olla, S. (2003). On the sector condition and homogenization of diffusions with a Gaussian drift. J. Funct. Anal. 197 179–211.
  • [14] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120, 238. English version: Russian Math. Surveys 40 73–145 (1985).
  • [15] Kozma, G. and Tóth, B. Central limit theorem for random walks in divergence-free random drift field: $\mathscr{H}_{-1}$ suffices. First arXiv version. Available at arXiv:1411.4171v1.
  • [16] Kumagai, T. (2014). Random Walks on Disordered Media and Their Scaling Limits. Lecture Notes in Math. 2101. Springer, New York.
  • [17] Löwner, K. (1934). Über monotone Matrixfunktionen. Math. Z. 38 177–216.
  • [18] Morris, B. and Peres, Y. (2005). Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 245–266.
  • [19] Oelschläger, K. (1988). Homogenization of a diffusion process in a divergence-free random field. Ann. Probab. 16 1084–1126.
  • [20] Olla, S. (2001). Central limit theorems for tagged particles and for diffusions in random environment. In Milieux Aléatoires (F. Comets and É. Pardoux, eds.). Panor. Synthèses 12 75–100. Soc. Math. France, Paris.
  • [21] Osada, H. (1983). Homogenization of diffusion processes with random stationary coefficients. In Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Math. 1021 507–517. Springer, Berlin.
  • [22] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979) (J. Fritz, D. Szász and J. L. Lebowitz, eds.). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
  • [23] Peled, R. (2017). High-dimensional Lipschitz functions are typically flat. Ann. Probab. 45 1351–1447.
  • [24] Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics. Vols. 1, 2. Academic Press, New York.
  • [25] Sethuraman, S., Varadhan, S. R. S. and Yau, H.-T. (2000). Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Comm. Pure Appl. Math. 53 972–1006.
  • [26] Tarrès, P., Tóth, B. and Valkó, B. (2012). Diffusivity bounds for 1d Brownian polymers. Ann. Probab. 40 695–713.
  • [27] Tóth, B. (1986). Persistent random walks in random environment. Probab. Theory Related Fields 71 615–625.
  • [28] Tóth, B. and Valkó, B. (2012). Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in $d=2$. J. Stat. Phys. 147 113–131.
  • [29] Varadhan, S. R. S. (1995). Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 31 273–285.
  • [30] Zeitouni, O. (2004). Lecture notes on random walks in random environment. In Lectures on Probability Theory and Statistics—Saint-Flour 2001 (J. Picard, ed.). Lecture Notes in Math. 1837. Springer, Berlin.