Characteristics of a random walk on a self-inflating support

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Self-similar dynamical processes are characterized by a growing length scale ξ which increases with time as ξ t¹/ᶻ, where z is the dynamical exponent. The best known example is a simple random walk with z=2. Usually such processes are assumed to take place on a static background. In this paper we address the question what changes if the background itself evolves dynamically. As an example we consider a random walk on an isotropically and homogeneously inflating space. For an exponentially fast expansion it turns out that the self-similar properties of the random walk are destroyed. For an inflation with power-law characteristics, however, self-similarity is preserved provided that the exponent controlling the growth is small enough. The resulting probability distribution is analyzed in terms of cumulant ratios. Moreover, the dynamical exponent z is found to change continuously with the control exponent.