Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates

One-dimensional run-and-tumble processes may converge towards some localized non-equilibrium steady state when the two velocities and/or switching rates are space-dependent. A long dynamical trajectory can be then analyzed via large deviations at Level 2.5 for joint probability of empirical densities, spatial currents and flows. The 2 densities alone derived optimization over More generally, any time-additive observable also obtained contraction from 2.5, or equivalently deformed generator method corresponding Doob conditioned process. Finally, intervals between consecutive events introduction alternate Markov chain that governs series all a trajectory.