Branching Processes

Large-time asymptotics for the density of a branching Wiener process

“…. , β (j ) ; t) are polynomials in t. Hence, by our induction hypothesis, E(U α (1) ,...,α (p) (t)) = m p E(U α (1) ,...,α (p) (t − 1)) + R α (1) ,...,α (p) (t), with |R α (1) ,...,α (p) (t)| ≤ ct β m (p−1)(t−1) for some β, c < ∞ independent of t. Iterating this completes the proof of our lemma for k = p and, hence, by induction, for all k.…”

“…The basic idea of the proof of (3.3) is that the subtraction eliminates the highest-order term in the expectation, leaving only sums of terms of the form U α (1) ,...,α (k) …”

“…. , β (j ) ; t) are polynomials in t and β (1) ,...,β (j ) is a finite sum. We next observe that = U β (1) ,...,β (j ) ,γ (1) ,...,γ (k) (t − 1)…”

“…. , γ (j ) ; t) are polynomials in t and γ (1) ,...,γ (j ) is a finite sum. Substituting this into (3.4) and using the fact that p k=0 p k (−1) k = 0, we find that the factors of m p U α (1) ,...,α (p) (t − 1) cancel, and we can write E({V α (t) − mV α (t − 1)} p ) = p−1 j =1 γ (1) ,...,γ (j ) g(α; n; γ (1) , .…”

An asymptotic maximum principle for essentially linear evolution models

“…(See [2] or [32] for the general context of multitype branching processes, and [26] for the application to mutation-selection models.) Important first questions concern the asymptotic properties of the systems discussed.…”