Positivity 4: 41–100, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Branching Random Walk in a Catalytic Medium.
I. Basic Equations
and LEONID V. BOGACHEV
Institut für Angewandte Mathematik, Universität Bonn, D-53115 Bonn, Germany.
Faculty of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia.
(Accepted 8 February 1999)
Abstract. We consider a continuous-time branching random walk on the integer lattice Z
with a ﬁnite number of branching sources, or catalysts. The random walk is assumed to be spatially
homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the
random walk, is governed by a Markov branching process. The quantities of interest are the local
numbers of particles (at each site) and the total population size. In the present paper, we derive
and analyze the Kolmogorov type backward equations for the corresponding Laplace generating
functions and also for the successive integer moments and the process extinction probability. In
particular, existence and uniqueness theorems are proved and the problem of explosion is studied
in some detail. We then rewrite these equations in the form of integral equations of renewal type,
which may serve as a convenient tool for the study of the process long-time behavior. The paper also
provides a technical foundation to some results published before without detailed proofs.
AMS subject classiﬁcations: Primary 60J80, 60J15; Secondary 39A70, 35R10, 60K05
Key words: branching random walk, catalysts, Laplace generating function, moments, backward
equations, difference operator, Cauchy problem in Banach space, renewal equations, explosion,
THE AIMS AND OUTLINE OF THE PAPER
In the present paper, we consider a continuous-time branching random walk (BRW)
on the integer lattice Z
(d 1) with ﬁnitely many sources, or catalysts, i.e., the
sites where the branching may occur. Informally, such a process can be described
as follows. Suppose that initially there is a single particle at site x ∈ Z
performs a random walk (RW) on Z
until reaching one of the catalysts located
at sites x
. Here, independently of the RW law, the particle undergoes a
branching process (BP), giving birth to a random offspring. In case no branching
has occurred, the particle continues its walk until the next return to a catalyst, and