Discrete-space and -time analog of a super-diffusive fractional Brownian motion

We discuss how to construct reliably well “a lattice and an integer time” version of super-diffusive continuous-space and -time fractional Brownian motion (fBm)—an experimentally relevant non-Markovian Gaussian stochastic process with an everlasting power-law memory on the time-evolution of thermal noises extending over the entire past. We propose two algorithms, which are both validated by extensive numerical simulations showing that the ensuing lattice random walks have not only the same power-law covariance function as the standard fBm, but also individual trajectories follow those of the super-diffusive fBm. Finding a lattice and an integer time analog of sub-diffusion fBm, which is an anti-persistent process, remains a challenging open problem. Our results also clarify the relevant difference between sub-diffusive and super-diffusive fBm, which are frequently seen as two very analogous realizations of processes with memory. They are indeed substantially different.