The Spread of a Catalytic Branching Random Walk

We investigate the total progeny of the killed branching random walk and give its precise tail distribution both in the critical and subcritical cases, which solves an open problem

According to Madaule [25], and A¨ıd´ekon, Berestycki, Brunet and Shi [3], Arguin, Bovier and Kistler [6] (for the branching brownian motion), the branching random walk seen from

For the critical branching random walk in Z 4 with branching at the origin only we find the asymptotic behavior of the probability of the event that there are particles at the origin

Let M n be the minimal position in the n-th generation, of a real-valued branching random walk in the boundary case.. This problem is closely related to the small deviations of a

We then introduce some estimates for the asymptotic behaviour of random walks conditioned to stay below a line, and prove their extension to a generalized random walk where the law

To complete the proof of Theorem 5.1, we first observe that the genealogy of particles close to the minimal displacement at time n in the branching random walk are either

We were able to obtain the asymptotic of the maximal displacement up to a term of order 1 by choosing a bended boundary for the study of the branching random walk, a method already

Branching processes with selection of the N rightmost individuals have been the subject of recent studies [7, 10, 12], and several conjectures on this processes remain open, such as