Greven A (1985)
Publication Type: Other publication type
Publication year: 1985
Publisher: Institute of Mathematical Statistics (IMS)
Book Volume: 13
Pages Range: 1133-1147
Journal Issue: 4
We consider a Markov process (rq ),eR + on (N)s (S= Z- ) with initial
distribution [ and the following time evolution: At rate bEyq(y, x)(y) a
particle is born at site x; at rate dr(x) a particle dies at site x. All particles
perform independent from each other continuous time random walk with
kernel p(x, y) and rate m. All particles at a site x die at rate D(x). Here
D(x) are random variables taking the values DI, D2 .(D2 2 DI O). We
assume {D(x)}Jx es to be stationary and ergodic. This paper studies the
features of the model for p(x, y), q(x, y) symmetric.
We calculate the exponential growth rate X of E(, q(x)) (with E denoting
conditional expectation with respect to the environment) and show that X is
nonrandom and strictly bigger than b - d - E(D(x)), if D2 > DI. We have
X = b - d - DI.
Introduce the process ( R )tER+ by setting it (x) (E(?t(x))) ? (x). A
critical phenomenon with respect to the parameter p = DI(d + ED(x))l
occurs in the sense that for p > p(2) the quantity E(g(x))2 grows exponen-
tially fast, while for p < p(2), X > 0 the exponential growth rate of E(nt(x))2
is 0. p(2) is the same as for a system with D(x) - DI and can be calculated
explicitly.
APA:
Greven, A. (1985). The coupled branching process in random environment. Institute of Mathematical Statistics (IMS).
MLA:
Greven, Andreas. The coupled branching process in random environment. Institute of Mathematical Statistics (IMS), 1985.
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