= Exponential cardinality supermartingale for the contact process on a tree
{title2=$\nu_\rho(A)=\rho^{|A|}$}
Consider the <Contact process> started from a finite infected set $A$ on a degree-$(d+1)$ <regular tree>, with $d\geq2$, recovery rate one and per-oriented-edge infection rate $\lambda>0$. Choose $0<\rho<1$ satisfying $\rho\lambda(d-1)\geq1$. The bounded functional $f(A)=\rho^{|A|}$ has generator
$$
Lf(A)=(1-\rho)\rho^{|A|}\left(\frac{|A|}{\rho}-\lambda|\partial_EA|\right).
$$
Recoveries decrease the infected count by one, whereas infections across the <edge boundary> increase it by one; summing these changes gives the formula.
A finite nonempty set induces a forest. If it has $c(A)$ <connected component>[connected components], <edge> counting gives $|\partial_EA|=(d-1)|A|+2c(A)$. Therefore the stated parameter condition makes $Lf\leq0$. At the absorbing empty state its generator is zero. Localizing the finite-particle process and using boundedness yields that $\rho^{|A_t|}$ is a <supermartingale>.
On extinction by time $t$, its value is one, so $\mathbb P_A(T_\varnothing\leq t)\leq\mathbb E_A\rho^{|A_t|}\leq\rho^{|A|}$. Letting $t\to\infty$ proves the extinction bound $\mathbb P_A(T_\varnothing<\infty)\leq\rho^{|A|}$. A strict choice $\rho<1$ exists when $\lambda(d-1)>1$, giving a sufficient survival condition from every nonempty finite seed. The range of $\rho$ is essential to both boundedness and the sign argument.
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