  What is the long-time behavior of the law of a contact process started with a
single infected site, distributed according to counting measure on the lattice?
This question is related to the configuration as seen from a typical infected
site and gives rise to the definition of so-called eigenmeasures, which are
possibly infinite measures on the set of nonempty configurations that are
preserved under the dynamics up to a multiplicative constant. In this paper, we
study eigenmeasures of contact processes on general countable groups in the
subcritical regime. We prove that in this regime, the process has a unique
spatially homogeneous eigenmeasure. As an application, we show that the
exponential growth rate is continuously differentiable and strictly decreasing
as a function of the recovery rate, and we give a formula for the derivative in
terms of the eigenmeasures of the contact process and its dual.
