Strong and weak convergence of the population size in a supercritical catalytic branching process

Abstract

A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for phase of pure global survival and strong local survival. Under wide conditions the limit theorems for the normalized total and local particles numbers in supercritical CBP are proved in the sense of almost surely convergence as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well. In the proofs the main role is played by recent results by the author devoted to classification of CBP and the moment analysis of the total and local particles numbers in CBP.

Spread of a catalytic branching random walk on a multidimensional lattice

“…A lot of them analyze asymptotic behavior of total and local particles numbers as time tends to infinity and only few investigate the spread of CBRW. Analysis of the mean total and local particles numbers implemented in the most general form in [10] as well as the strong and weak limit theorems established in [11] shows that CBRW can be classified as supercritical, critical and subcritical like ordinary branching processes and only in the supercritical regime the total and local particles numbers grow jointly to infinity. For this reason, it is of primary interest to consider spread of particles population in supercritical CBRW.…”

Fluctuations of the Propagation Front of a Catalytic Branching Walk

“…x ∈ Z d , follows from Theorem 4 in paper [11]. It means that we obtain a complete description of the fluctuations of the propagation front of the particles population in CBRW under its local non-degeneracy.…”

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

“…Moreover, as stated in Theorem 4 of [9], both total and local particles numbers, being normalized by their means, converge in distribution to a non-degenerate random variable which vanishes with probability Q(x) of local extinction of population in CBRW starting at the point x. Similarly, as shows Theorem 1, the trivial relation M t /L t → 0, as t → ∞, is only realized with the same probability Q(x) of local extinction of population in CBRW with starting point x.…”

“…Recall the definition of the local extinction probability Q(x, y) := P x (lim sup t→∞ µ(t; y) = 0), x, y ∈ Z, where µ(t; y) is the number of particles in CBRW at point y at time t (local particles number). Theorem 2 in [9] asserts that the function Q(x, y) depends on x only, i.e. Q(x, y) = Q(x), and satisfies some system of algebraic equations provided there.…”

“…The function ϕ has already emerged in Theorem 4 in [9] as the Laplace transform of the limiting distribution arising in the limit theorem for total and local particles numbers in CBRW. Thus, to obtain a known limit in Theorem 1 we consider a variable λ −1/γ instead of λ (and M t /L t instead of (M t /L t ) −γ , since the latter expression is undefined when M t < 0).…”