Relation between stochastic processes and thermodynamics of trajectories

The process of fluctuations of trajectory observables of stochastic systems is associated with processes with independent increments from the risk theory. Boundary problems are considered: first-passage times (when a certain value of the variable is first reached), and overshoots at the moments of the first-passage times to get positive and negative levels. A correspondence between the expressions of the theory of random processes and thermodynamics of trajectories is established, as well as deviations from such correspondence for the process of fluctuations of trajectory observables. Relations between the general regularities of the first-passage times in the theory of random processes and the 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 the approaches of the theory of random processes and statistical physics are indicated.