Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

Abstract

For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a nontrivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables.

Fluctuations of the Propagation Front of a Catalytic Branching Walk

“…where G * 1 (ν) = β 1 / (ν + β 1 ). Hence, taking into account definition (13) and operating similarly to deriving the asymptotic behavior of the function J 3 (t), as t → ∞, we get relation (25). Lemma 3 is proved completely.…”

“…of the first hitting time of point w k by the random walk S under the taboo on states W k , when the starting point of the walk S is x (for hitting times under taboo, see, e.g., [9]). According to Lemma 7 of paper [13], under conditions (1) and (2) the system (8) has a unique solution ϕ( · ; w j ), j = 1, . .…”

“…The following lemma proved in [13] contains an integral equation for the probability E(t; u). Lemma 1 Let condition (1) be valid.…”

“…According to Lemma 7 of paper [13], under conditions ( 1) and (2) the system (8) has a unique solution ϕ( • ; w j ), j = 1, . .…”

“…The proof of Lemma 4 mainly resembles the proof of the previous Lemma 3. Firstly, once again apply the estimate 0 ≤ I(t; u) ≤ P 0 (S(t) > u), t, u ≥ 0, following from the proof of Lemmas 1 and 2 in paper [13] under condition (1). Secondly, use the following representation…”