  We consider a general honest homogeneous continuous-time Markov process with
restarts. The process is forced to restart from a given distribution at time
moments generated by an independent Poisson process. The motivation to study
such processes comes from modeling human and animal mobility patterns, restart
processes in communication protocols, and from application of restarting random
walks in information retrieval. We provide a connection between the transition
probability functions of the original Markov process and the modified process
with restarts. We give closed-form expressions for the invariant probability
measure of the modified process. When the process evolves on the Euclidean
space there is also a closed-form expression for the moments of the modified
process. We show that the modified process is always positive Harris recurrent
and exponentially ergodic with the index equal to (or bigger than) the rate of
restarts. Finally, we illustrate the general results by the standard and
geometric Brownian motions.
