Markov Processes with Restart

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.

💡 Research Summary

The paper introduces and rigorously analyzes a continuous‑time homogeneous Markov process that is periodically forced to restart from a prescribed distribution. Restart times are generated by an independent Poisson process with rate λ, and at each restart the state is sampled from a fixed probability measure ν on the state space E. The authors first derive an explicit relationship between the transition kernel P(t, x,·) of the original process and the transition kernel Q(t, x,·) of the restarted process:

 Q(t, x, A) = e^{‑λt} P(t, x, A) + λ ∫₀^{t} e^{‑λs} ∫_{E} P(s, y, A) ν(dy) ds.

This formula shows that the restarted dynamics are a convex combination of the original dynamics (weighted by the probability that no restart has occurred up to time t) and a mixture of the original dynamics started from the restart distribution ν (weighted by the probability that a restart occurred at time s ≤ t).

Using this representation, the authors obtain a closed‑form expression for the invariant probability measure μ* of the restarted process:

 μ*(A) = λ ∫₀^{∞} e^{‑λs} ∫_{E} P(s, y, A) ν(dy) ds.

When ν coincides with an invariant measure of the original process, μ* reduces to that same measure, confirming that the restart mechanism does not disturb an already stationary regime. Otherwise, μ* is a weighted average of the original transition probabilities started from ν, with exponential weighting determined by λ.

For processes evolving on ℝ^d, the authors further derive explicit formulas for moments of the restarted process. The first moment, for example, satisfies

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