  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.
