We study the ergodic behavior of the contact process on infinite
connected graphs of bounded degree. We show that the fundamental notion of
complete convergence is not as well behaved as it was thought to be. In
particular there are graphs for which complete convergence holds in any number
of separated intervals of values of the infection parameter and fails for the
other values of this parameter. We then introduce a basic invariant probability
measure related to the recurrence properties of the process, and an associated
notion of convergence that we call “partial convergence.” This
notion is shown to be better behaved than complete convergence, and to hold in
certain cases in which complete convergence fails. Relations between partial
and complete convergence are presented, as well as tools to verify when these
properties hold. For homogeneous graphs we show that whenever recurrence takes
place (i.e., whenever local survival occurs) there are exactly two extremal
invariant measures.
@article{1023481114,
author = {Salzano, Marcia and Schonmann, Roberto H.},
title = {The second lowest extremal invariant measure of the contact
process},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 1846-1871},
language = {en},
url = {http://dml.mathdoc.fr/item/1023481114}
}
Salzano, Marcia; Schonmann, Roberto H. The second lowest extremal invariant measure of the contact
process. Ann. Probab., Tome 25 (1997) no. 4, pp. 1846-1871. http://gdmltest.u-ga.fr/item/1023481114/