Stochastic walker with variable long jumps

Motivated by recent interest in the stochastic resetting of a random walker, we propose a generalized model where the random walker takes stochastic jumps of lengths proportional to its present position with certain probability, otherwise it makes forward and backward jumps of fixed (unit) length with given rates. The model exhibits a rich stochastic dynamic behavior. We obtain exact analytic results for the first two moments of the walker's displacement and show that a phase transition from a diffusive to superdiffusive regime occurs if the stochastic jumps of lengths that are twice (or more) of its present positions are allowed. This phase transition is accompanied by a reentrant diffusive behavior.

Automated summary

Motivated by recent interest in stochastic resetting of a random walker, a generalized model is proposed in which the walker takes stochastic jumps of lengths proportional to its current position with certain probability. The model reveals rich stochastic dynamic behavior and a phase transition from a diffusive to a superdiffusive regime if the jumps of lengths that are twice (or more) of its current positions are allowed. This phase transition is accompanied by a reentrant diffusive behavior.