= Exponential cardinality supermartingale for the contact process on a tree
{title2=$\nu_\rho(A)=\rho^{|A|}$}
For the <contact process> with per-<edge> infection rate $\lambda$ and recovery rate one on a degree-$(d+1)$ <regular tree>, the <infinitesimal generator> gives $L\nu_\rho(A)=(1-\rho)\nu_\rho(A)(|A|/\rho-\lambda|\partial_E A|)$. The <edge boundary of a finite forest in a regular tree> implies $L\nu_\rho\le0$ whenever $\rho\lambda(d-1)\ge1$, including the absorbing empty state where the generator is zero. Thus the displayed functional is a bounded <supermartingale>, and the extinction <probability> from $A$ is at most $\rho^{|A|}$.
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