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First passage under restart with branching is proposed as a generalization of first passage under restart. Strong motivation to study this generalization comes from the observation that restart with branching can expedite the completion of processes that cannot be expedited with simple restart; yet a sharp and quantitative formulation of this statement is still lacking. We develop a comprehensive theory of first passage under restart with branching. This reveals that two widely applied measures of statistical dispersion-the coefficient of variation and the Gini index-come together to determine how restart with branching affects the mean completion time of an arbitrary stochastic process. The universality of this result is demonstrated and its connection to extreme value theory is also pointed out and explored. arXiv:1807.09363v3 [cond-mat.stat-mech]

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target site. The agents performing the search combine local scanning-conducted by a 1D motion along the strandand random relocations on the strand-conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search durations are analysed, and their limiting probability distributions-for long DNA strands-are obtained in closed form. The results obtained (i) encompass the cases of single, parallel and massively parallel searches, taking place in the presence of either finite-mean or heavytailed relocation times, (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar, and sub-diffusive motions, (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism, and (iv) facilitate the derivation of optimal relocation strategies.

Restarting a deterministic process always impedes its completion. However, it is known that restarting a random process can also lead to an opposite outcome-expediting completion. Hence, the effect of restart is contingent on the underlying statistical heterogeneity of the process' completion times. To quantify this heterogeneity we bring a novel approach to restart: the methodology of inequality indices, which is widely applied in economics and in the social sciences to measure income and wealth disparity. Using this approach we establish an 'inequality roadmap' for the mean-performance of sharp restart: a whole new set of universal inequality criteria that determine when restart with sharp timers (i.e. with fixed deterministic timers) decreases/increases mean completion. The criteria are based on a host of inequality indices including Bonferroni, Gini, Pietra, and other Lorenzcurve indices; each index captures a different angle of the restart-inequality interplay. Utilizing the fact that sharp restart can match the mean-performance of any general restart protocol, we prove-with unprecedented precision and resolutionthe validity of the following statement: restart impedes/expedites mean completion when the underlying statistical heterogeneity is low/high.