Relation between stochastic processes and thermodynamics of trajectories

The process of fluctuations of trajectory observables of stochastic systems is related to processes with independent increments from risk theory. The first-passage times of variables of the thermodynamics of trajectories, in particular, dynamic activity, are considered. A correspondence between expressions of the theory of random processes and thermodynamics of trajectories, as well as deviations from such a correspondence for the process of fluctuations of trajectory observables, are established. The connections between general regularities of first-passage times in the theory of random processes and thermodynamics of trajectories are discussed. A more complete use of the theory of random processes in physical problems is proposed, and the possibilities of combining approaches of the theory of random processes and statistical physics are indicated.