Asymptotic properties of a bold random walk

Abstract

In a recent paper we proposed a non-Markovian random walk model with memory of the maximum distance ever reached from the starting point (home). The behavior of the walker is at variance with respect to the simple symmetric random walk (SSRW) only when she is at this maximum distance, where, having the choice to move either farther or closer, she decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold and her behavior turns out to be super-diffusive, otherwise she is timorous and her behavior turns out to be sub-diffusive. The scaling behavior vary continuously from sub-diffusive (timorous) to super-diffusive (bold) according to a single parameter $\gamma \in R$. We investigate here the asymptotic properties of the bold case in the non ballistic region $\gamma \in [0,1/2]$, a problem which was left partially unsolved in \cite{S}. The exact results proved in this paper require new probabilistic tools which rely on the construction of appropriate martingales of the random walk and its hitting times.

References in corpus (8)

• From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion
• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• On the strong anomalous diffusion
• Random walks with preferential relocations to places visited in the past and their application to biology
• Simple stochastic models showing strong anomalous diffusion
• On anomalous diffusion and the out of equilibrium response function in one-dimensional models
• Molecular theory of anomalous diffusion
• Asymptotic analysis of a random walk with a history-dependent step length

Cited by in corpus (1)

• Linear and anomalous front propagation in system with non Gaussian diffusion: the importance of tails