Large deviations principle for a stochastic process with random reinforced relocations

Stochastic processes with random reinforced relocations have been introduced in a series of papers by Boyer and co-authors (Boyer and Solis Salas 2014, Boyer and Pineda 2016, Boyer, Evans and Majumdar 2017) to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some “memory kernel”, and jumps to its value at that random time. We prove a quenched large deviations principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markov because of the relocations. Furthermore, the random inter-relocation times act as a random environment.