Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called level 2.5’') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.