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{title2=$\underline\nu=\delta_0,\quad\overline\nu=\lim_{t\to\infty}\delta_1P_t$}

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 $\delta_1P_t$ decrease in <stochastic domination> as $t$ grows. Their <weak convergence of probability measures>[weak limit] $\overline\nu$ exists and is invariant by the <Feller semigroup>[Feller property].

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

Self-duality identifies the one-site <probability density function>[density] with survival from that site:
$$
\overline\nu(\eta(x)=1)=\mathbb P^{\{x\}}(A_t\ne\varnothing\text{ for every }t)=:\theta_x(\lambda).
$$
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 probability measure of a Markov process>[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.<Contact Process by Rick Durrett>[Theorems 1–2 and the single-site consequence of duality]{ref}

Extremal invariant measures of the contact process

Invariant probability law of a Markov process

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 37 / 4 / a / Solution

Extremal invariant measures of the contact process ( $\underline{ν} = δ_{0}, \overline{ν} = lim_{t \to \infty} δ_{1} P_{t}$ )

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Extremal invariant measures of the contact process (ν​=δ0​,ν=limt→∞​δ1​Pt​)

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Extremal invariant measures of the contact process ( $\underline{ν} = δ_{0}, \overline{ν} = lim_{t \to \infty} δ_{1} P_{t}$ )

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