Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory

Abstract: We analyse large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_t of the time-averaged quantity. First we consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behaviour, which differ from the Markovian case. In particular, the memory can lead to large deviation principles with reduced speeds, and to non-analytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian symmetric exclusion process.