It is known that the contact process on a homogeneous tree of degree
$d+1\geq3$ has a weak survival phase, in which the infection survives
with positive probability but nevertheless eventually vacates every finite
subset of the tree. It is shown in this paper that in the weak survival phase
there exists a spherically symmetric invariant measure whose density decays
exponentially at infinity, thus confirming a conjecture of Liggett. The proof
is based on a study of the relationships between various thermodynamic
parameters and functions associated with the contact process initiated by a
single infected site. These include (1) the growth profile, which
determines the exponential rate of growth in space-time on the event of
survival, (2) the exponential rate $/beta$ of decay of the hitting probability
function at infinity (also studied by the author) and (3) the exponential rate
$\beta$ of decay in time $t$ of the probability that the initial infected
site is infected at time $t$. It is shown that $\beta$ is a strictly
increasing function of the infection rate $\lambda$ in the weak survival phase,
and that $\beta = 1/\sqrt{d}$ at the upper critical point $\lambda_2$
demarcating the boundary between the weak 2 and strong survival phases. It is
also shown that $\eta < 1$ except at $\lambda_2$, where $\eta =1$.