Stochastic processes for modeling and evaluating atomic clock behaviour

The behavior of the atomic clocks is typically studied considering their phase and frequency error with respect to an ideal clock. This quantity is known to be well modeled by a stochastic process. We presented and study the problem of the first passage time of the stochastic processes across two fixed constant barriers. This topic is also known as survival probability between two absorbing barriers and it can be studied by means of the infinitesimal generator. We present the application of these results to atomic clocks modelling, in particular to the prediction of their errors and to the evaluation of the probability that the clock error exceeds the limit of permissible error a certain time after synchronization. This aspect has an interesting application in the frame of the Mutual Recognition Arrangement where it may be important not only demonstrating that two standards are “equivalent”, but also evaluating how long such equivalence can last.