The Annals of Probability

Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension

L. R. G. Fontes, M. Isopi, and C. M. Newman

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Abstract

Let $\tau = (\tau_i : i \in \mathbb{Z})$ denote i.i.d. positive random variables with common distribution $F$ and (conditional on $\tau$) let $X = )X_t : t \geq 0, X_0 = 0)$, be a continuous-time simple symmetric random walk on $\mathbf{Z}$ with inhomogeneous rates $(\tau_i^{-1} : i \in \mathbb{Z})$. When $F$ is in the domain of attraction of a stable law of exponent $\alpha < 1$ [so that $\mathbb{E}(\tau_i) = \infty$ and $X$ is subdiffusive], we prove that $(X, \tau)$, suitably rescaled (in space and time), converges to a natural (singular) diffusion $Z = (Z_t : t \geq 0, Z_0 = 0)$ with a random (discrete) speed measure $\rho$. The convergence is such that the “amount of localization,” $\mathbb{E} \sum_{i \in \mathbb{Z}}[\mathbb{P}(X_t = i|\tau)]^2$ converges as $t \to \infty$ to $\mathbb{E} \sum_{z \in \mathbb{R}}[\mathbb{P}(Z_s = z|\rho)]^2 > 0$, which is independent of $s > 0$ because of scaling/self-similarity properties of $(Z, \rho)$. The scaling properties of $(Z, \rho)$ are also closely related to the “aging” of $(X, \tau)$. Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks $Y^{(\epsilon)}$ with (nonrandom) speed measures $\mu^{(\epsilon)} \to \mu$ (in a sufficiently strong sense).

Article information

Source
Ann. Probab., Volume 30, Number 2 (2002), 579-604.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481003

Digital Object Identifier
doi:10.1214/aop/1023481003

Mathematical Reviews number (MathSciNet)
MR1905852

Zentralblatt MATH identifier
1015.60099

Subjects
Primary: 60K37: Processes in random environments 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60G18: Self-similar processes
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
aging localization quasidiffusions disordered systems scaling limits random walks in random environments self-similarity

Citation

Fontes, L. R. G.; Isopi, M.; Newman, C. M. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2002), no. 2, 579--604. doi:10.1214/aop/1023481003. https://projecteuclid.org/euclid.aop/1023481003


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References

  • [1] FONTES, L. R., ISOPI, M. and NEWMAN, C. M. (1999). Chaotic time dependence in a disordered spin system. Probab. Theory Related Fields 115 417-443.
  • [2] BOUCHAUD, J.-P., CUGLIANDOLO, L., KURCHAN, J. and MÉZARD, M. (1998). Out of equilibrium dynamics in spin-glasses and other glassy systems. In Spin-glasses and Random Fields (A. P. Young, ed.). World Scientific, Singapore.
  • [3] STONE, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638-660.
  • [4] HAVLIN, S. and BEN-AVRAHAM, D. (1987). Diffusion in disordered media. Adv. Phys. 36 695-798.
  • [5] ISICHENKO, M. B. (1992). Percolation, statistical topography and transport in random media. Rev. Modern Phys. 64 961-1043.
  • [6] SINAI, YA. G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 256-268.
  • [7] SOLOMON, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31.
  • [8] GOLOSOV, A. O. (1984). Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 491-506.
  • [9] LE DOUSSAL, P., MONTHUS, C. and FISHER, D. S. (1999). Random walks in onedimensional random environments: exact renormalization group analysis. Phys. Rev. E 59 1795-1810.
  • [10] DEMBO, A., GUIONNET, A. and ZEITOUNI, O. (2001). Aging properties of Sinai's model of random walk in random environment. Available at arXiv.org/pdf/math.PR/0105215.
  • [11] KAWAZU, K. and KESTEN, H. (1984). On birth and death processes in symmetric random environment. J. Statist. Phys. 37 561-576.
  • [12] SCHUMACHER, S. (1984). Diffusions with random coefficients. Ph.D. dissertation, Univ. California, Los Angeles.
  • [13] SCHUMACHER, S. (1985). Diffusions with random coefficients. In Particle Systems, Random Media and Large Deviations (R. Durrett, ed.) 351-356. Amer. Math. Soc., Providence, RI.
  • [14] BEN AROUS, G., DEMBO, A. and GUIONNET, A. (2001). Aging of spherical spin glasses. Probab. Theory Related Fields 120 1-67.
  • [15] NEWMAN, C. M. and STEIN, D. L. (1999). Equilibrium pure states and nonequilibrium chaos. J. Statist. Phys. 94 709-722.
  • [16] NANDA, S., NEWMAN, C. M. and STEIN, D. L. (2000). Dynamics of Ising spin systems at zero temperature. In On Dobrushin's Way. From Probability Theory to Statistical Physics (R. Minlos, S. Shlosman and Y. Suhov, eds.) 183-194. Amer. Math. Soc., Providence, RI.
  • [17] ITÔ, K. and MCKEAN, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, New York.
  • [18] RINN, B., MAASS, P. and BOUCHAUD, J.-P. (2000). Multiple scaling regimes in simple aging models. Phys. Rev. Lett. 84 5403-5406.
  • [19] 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.
  • [20] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications II. Wiley, New York.
  • [21] KÜCHLER, U. (1980). Some asymptotic properties of the transition densities of onedimensional quasidiffusions. Publ. Res. Inst. Math. Sci. 16 245-268.
  • [22] KNIGHT, F. B. (1981). Characterization of the Levy measures of inverse local times of gap diffusion. Progr. Probab. Statist. 1 53-78.
  • [23] KOTANI, S. and WATANABE, S. (1982). Kre in's spectral theory of strings and generalized diffusion processes. Lecture Notes in Math. 923 235-259. Springer, New York.
  • [24] RESNICK, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • [25] SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
  • [26] SATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • NEW YORK, NEW YORK 10012 E-MAIL: newman@cims.nyu.edu