Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Abstract

Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper for the offspring number in subcritical regime finiteness of the moment of order 1 + δ is required where δ is some positive number.

Spread of a catalytic branching random walk on a multidimensional lattice

“…Theorem 1 means that if we divide the position coordinates of each particle existing in CBRW at time t by t and then let t tend to infinity, then in the limit there are a.s. no particles outside the set P ∪ Q and under condition of infinite number of visits of catalysts there are a.s. particles on P. In this sense it is natural to call the border P the propagation front of the particles population. The following theorem refines assertion (7) of Theorem 1 and states that each point of P can be considered as a limiting point for the normalized particles positions in CBRW.…”

“…Combination of the proved part of Theorem 1 and formula (17) implies the assertion of Theorem 2 for the case of a single catalyst at 0 and the starting point 0 whenever Eξ 2 1 < ∞. Under the same conditions statement (7) of Theorem 1 is established since relation (17) entails P 0 (ω : ∃t 0 (ω) such that ∀t ≥ t 0 (ω) one has ∃v ∈ N (t), X v (t) / ∈ tQ ε | I) = 1, for each ε ∈ (0, ν).…”

“…Moreover, we concentrate on the case Eξ 2 1 < ∞. Let us establish Theorem 2 and statement (7) of Theorem 1 under these assumptions.…”

“…Study of different variants of CBRW goes back to more than 10 years (see, e.g., [1] and [30]), although most of papers in this research domain have been published recently, see, for instance, [31], [21], [27], [7], [16], [8] and [14]. 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.…”

Complete Classification of Catalytic Branching Processes

“…For instance, even for critical symmetric branching random walk on Z d with a single catalyst (see [2]) there are four different asymptotic formulae for m n (t; x, y) and M n (t; x) depending on dimension d = 1, 2, 3 or 4 and thereby on the decay rate of transition probabilities (note that for d ≥ 5 Theorem 1 generalizes the corresponding results in [2]). Moreover, for subcritical branching random walk on Z d with a single catalyst (see [7]) the decay orders of m n (t; x, y) do not coincide for different d ∈ N.…”