Large deviations principle for a stochastic process with random reinforced relocations

Stochastic processes with random reinforced relocations have been introduced in the physics literature 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. We also prove a partial large deviations principle for the occupation measure of the process in the case when the memory kernel is decreasing. We use a similar approach to prove large deviations principle for the profile of the weighted random recursive tree (WRRT) in the case when older nodes are more likely to attract new links than more recent nodes.