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For the Contact process on a countable undirected bounded-degree graph with recovery rate one and symmetric edge infection rate $\lambda$, the all-healthy law $\underline{\nu }={\delta }_0$ is invariant. The graphical representation preserves coordinatewise order. Since every configuration is below the all-infected configuration, the laws ${\delta }_1{P}_t$ decrease in stochastic domination as $t$ grows. Their weak limit $\overline{\nu }$ exists and is invariant by the Feller property.

Every invariant probability law lies between $\underline{\nu }$ and $\overline{\nu }$ in stochastic order. These are the least and greatest invariant laws; classifying all extremal invariant laws requires further hypotheses. The greatest law is also an extreme point: in a convex decomposition into invariant laws, both components are dominated by it, and equality of all increasing finite-cylinder expectations forces both to equal it.

Self-duality identifies the one-site density with survival from that site:On a transitive graph the right side is independent of $x$ and is conventionally written $\theta (\lambda )$. On a general graph its site dependence must be retained. The two bounding invariant laws coincide exactly when ${\theta }_x(\lambda )=0$ for every site: countability then forces the upper law to assign probability one to the empty configuration. The lattice version of the maximal-law and duality construction is given by the two basic contact-process theorems in Rick Durrett's survey.^[ref]