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

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.

51 pages

References in corpus (6)

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• Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments
• Microscopic concavity and fluctuation bounds in a class of deposition processes
• Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Moderate deviations for diffusion in time dependent random media
• Large deviations and wandering exponent for random walk in a dynamic beta environment
• Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction
• A quenched local limit theorem for stochastic flows
• KPZ equation limit of sticky Brownian motion
• Multiplicative SHE limit of random walks in space-time random environments
• Tiny fluctuations of the averaging process around its degenerate steady state