Preserving of Markovness whilst delayed reflection

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 157-174

Résumé

A one-dimensional locally-Markov diffusion process with positive range of values is considered. This process is assumed to be reflected from the point 0. All variants of reflection preserving the semi-Markov property are described. The reflected process prolongs to be locally-Markov in open intervals, but it can loose the global Markov property. The reflection is characterized by $\alpha(r)$ which is the first exit time from semi-interval $[0,r)$ after the first hitting time at 0 (for any $r>0$). A distribution of this time-interval is used for deriving a time change a process with instantaneous reflection into a process with delayed reflection. A process which preserves its markovness after the delayed reflection is proved to have a special distribution of the set of time points when the process has zero meaning during the time $\alpha(r)$. This discontinuum set has exponentially distributed Lebesgue measure.