= Extremal invariant measures of the contact process
{title2=$\underline\nu=\delta_0,\quad\overline\nu=\lim_{t\to\infty}\delta_1P_t$}
The <graphical representation of the contact process> preserves coordinatewise order. Its law from the all-infected state decreases in <stochastic domination> with time, and its limit is invariant by the <Feller semigroup> property. The all-healthy state is absorbing. Any invariant probability law lies between these two laws. <Duality of the contact process> gives $\overline\nu(\eta(x)=1)=\theta(\lambda)$ when the recovery rate is one. Consequently the two extremal laws coincide exactly when the <survival probability of the contact process> is zero.
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