Fluctuations of Biggins’ martingales at complex parameters

Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

“…Indeed, the function on R 2 that maps (x, y) to e − 1 ϑ x f K (x)y is continuous and vanishes for all sufficiently large x. Therefore, The second assertion can be proved similarly as in the proof of Theorem 2.5 in [17]. More precisely, it follows from the dominated convergence theorem once we have proved that…”

“…In both cases, if φ 0 satisfies (1.8) with γ > ϑ, then µ 0 ∈ M 1 p (R) for some ϑ < p ≤ 2. In this more general situation, we demonstrate how the asymptotic behavior of µ t can be derived from recent progress on kinetic-type equations [12] and the extrema of branching random walks [17,21]. Our proof works under a mild X log Xtype moment condition (cf.…”

“…Our main result, Theorem 1.3, follows directly from Lemma 2.1 and the following proposition. The bulk of the proof of this proposition can be adopted from the proof of Theorem 2.5 in [17], however at some points changes are needed. In what follows, we repeat the major steps of the proof of the cited theorem adjusted to the situation here and point out the changes that are required.…”

“…Proof. The lemma follows from (the proof of) Lemma 5.3 in [17], except at one point in the proof where the Topchiȋ-Vatutin inequality is used (Lemma A.1 in the cited reference). The use of the latter inequality has to be replaced by an application of (1.11) The rest of the proof carries over without changes.…”

“…Then f (x, y) Z * n (dx , dy) → f (x) Z * ∞ (dx , dy) for all bounded continuous functions f : R 2 → R satisfying f (x, y) = 0 for all sufficiently large x.Source. The lemma is a special case of Lemma 5.2 in[17]. Under the assumptions of Proposition 3.2, for any δ > 0 and any measurable…”