Geodesic
Asymptotic behavior of the mean number of particles of branching random walk on $\mathbf Z^d$ with periodic sources of branching
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 26, Tome 466 (2017), pp. 234-256

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We consider a continuous-time branching random walk on $\mathbf Z^d$ with birth and death of particles at a periodic set of points (the sources of branching). Spectral properties of an evolution operator of the mean number of particles are studied. We derive a representation of the mean value of particle number in a form of asymptotic series. 
@article{ZNSL_2017_466_a15,
     author = {M. V. Platonova and K. S. Ryadovkin},
     title = {Asymptotic behavior of the mean number of particles of branching random walk on $\mathbf Z^d$ with periodic sources of branching},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {234--256},
     publisher = {mathdoc},
     volume = {466},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a15/}
}
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M. V. Platonova; K. S. Ryadovkin. Asymptotic behavior of the mean number of particles of branching random walk on $\mathbf Z^d$ with periodic sources of branching. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 26, Tome 466 (2017), pp. 234-256. http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a15/