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The random average process and random walk in a space-time random environment in one dimension

Marton Balazs·Firas Rassoul-Agha·Timo Seppalainen
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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 equilibria, 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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Marton Balazs, Firas Rassoul-Agha, Timo Seppalainen. 2006-04-07. The random average process and random walk in a space-time random environment in one dimension. https://doi.org/10.1007/s00220-006-0036-y

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