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We consider a class of piecewise-deterministic Markov processes where thestate evolves according to a linear dynamical system. This continuous timeevolution is interspersed by discrete events that occur at random times andchange (reset) the state based on a linear affine map. In particular, weconsider two families of discrete events, with the first family of resetsoccurring at exponentially-distributed times. The second family of resets isgenerally-distributed, in the sense that, the time intervals between events areindependent and identically distributed random variables that follow anarbitrary continuous positively-valued probability density function. For thisclass of stochastic systems, we provide explicit conditions that lead to finitestationary moments, and the corresponding exact closed-form moment formulas.These results are illustrated on an example drawn from systems biology, where aprotein is expressed in bursts at exponentially-distributed time intervals,decays within the cell-cycle, and is randomly divided among daughter cells whengenerally-distributed cell-division events occur. Our analysis leads to novelresults for the mean and noise levels in protein copy numbers, and we decomposethe noise levels into components arising from stochastic expression, randomcell-cycle times, and partitioning. Interestingly, these individual noisecontributions behave differently as cell division times become more random. Insummary, the paper expands the class of stochastic hybrid systems for whichstatistical moments can be derived exactly without any approximations, andthese results have applications for studying random phenomena in diverse areas.

Linear piecewise-deterministic Markov processes with families of random discrete events