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The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

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Abstract

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n 1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n 1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.

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Authors and Affiliations

  1. Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, Madison, WI, 53706, USA

    Márton Balázs & Timo Seppäläinen

  2. Mathematical Biosciences Institute, Ohio State University, 231 West 18th Avenue, Columbus, OH, 43210, USA

    Firas Rassoul-Agha

Authors
  1. Márton Balázs
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  2. Firas Rassoul-Agha
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  3. Timo Seppäläinen
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Corresponding author

Correspondence to Timo Seppäläinen.

Additional information

Communicated by H. Spohn

M. Balázs was partially supported by Hungarian Scientific Research Fund (OTKA) grant T037685.

T. Seppäläinen was partially supported by National Science Foundation grant DMS-0402231.

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Balázs, M., Rassoul-Agha, F. & Seppäläinen, T. The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension. Commun. Math. Phys. 266, 499–545 (2006). https://doi.org/10.1007/s00220-006-0036-y

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