Dissipative Processes with Infinite Memory

We study the process of random growth of surfaces approximating it by fractional Brownian motion (FBM) with scaling index $H$. The diffusion trajectories generated by the ballistic deposition ($H=1/3$) and Edward-Wilkinson ($H=1/4$) models are analyzed and the distribution of time intervals between two consecutive origin re-crossings are calculated numerically. This distribution follows the inverse power-law, $\psi(\tau) \propto 1/{\tau}^{\mu}$. For pure FBM $\mu = 2-H$ if $1/3