Communications in Mathematical Physics

, Volume 266, Issue 2, pp 499–545 | Cite as

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

  • Márton Balázs
  • Firas Rassoul-Agha
  • Timo Seppäläinen


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.


Random Walk Gaussian Process Fractional Brownian Motion Random Environment Exclusion Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Márton Balázs
    • 1
  • Firas Rassoul-Agha
    • 2
  • Timo Seppäläinen
    • 1
  1. 1.Mathematics DepartmentUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Mathematical Biosciences InstituteOhio State UniversityColumbusUSA

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