Dynamical phase transitions in certain nonergodic stochastic processes

We present a class of stochastic processes in which the large deviation functions of time-integrated observables exhibit singularities that relate to dynamical phase transitions of trajectories. These illustrative examples include Brownian motion with a death rate or in the presence of an absorbing wall, for which we consider a set of empirical observables such as the net displacement, local time, residence time, and area under the trajectory. Using a backward Fokker-Planck approach, we derive the large deviation functions of these observables and demonstrate how singularities emerge from a competition between survival and diffusion. Furthermore, we analyze this scenario using an alternative approach with tilted operators, showing that at the singular point, the effective dynamics undergoes an abrupt transition. Extending this approach, we show that similar transitions may generically arise in Markov chains with transient states. This scenario is robust and generalizable for non-Markovian dynamics and for many-body systems, potentially leading to multiple dynamical phase transitions. We have confirmed most of our findings on the singular large-deviation function using rare-event simulation techniques.