A simple proof of exponential decay of subcritical contact processes

Abstract

This paper gives a new, simple proof of the known fact that for contact processes on general lattices, in the subcritical regime the expected number of infected sites decays exponentially fast as time tends to infinity. The proof also yields an explicit bound on the survival probability below the critical recovery rate, which shows that the critical exponent associated with this function is bounded from below by its mean-field value. The main idea of the proof is that if the expected number of infected sites decays slower than exponentially, then this implies the existence of a harmonic function that can be used to show that the process survives for any lower value of the recovery rate.

Appendix A: Transformation of submartingales

Let $\mathcal{S}$ be a countable set and let G be a so-called Q-matrix on $\mathcal{S}$, i.e., $(G(x,y))_{x,y\in \mathcal{S}}$ are real constants such that $G(x,y)\ge 0$ for $x\ne y$ and $\sum _{y\in \mathcal{S}}G(x,y)=0$. For any real function f on $\mathcal{S}$, we write

$$\begin{aligned} Gf(x):=\sum _{y\in \mathcal{S}}G(x,y)f(y) =\sum _{y\in \mathcal{S}}G(x,y)\big (f(y)-f(x)\big ) \quad (x\in \mathcal{S}), \end{aligned}$$

(32)

whenever the infinite sums are well-defined. Then G is the the generator of a (possibly explosive) continuous-time Markov chain $(X_t)_{t\ge 0}$ in $\mathcal{S}$. A function h such that $Gh\ge 0$ is called subharmonic. The following simple lemma says, roughly speaking, that an unbounded, nonnegative subharmonic function that has a sufficiently positive drift and not too large fluctuations can be transformed into a bounded subharmonic function.

Lemma 3

(Transformation of submartingales) Let h be a real function on $\mathcal{S}$ and let $\varepsilon >0$. Then the function

$$\begin{aligned} f_\varepsilon (x):=\frac{1}{\varepsilon }\big (1-e^{-\varepsilon h(x)}\big )\quad (x\in \mathcal{S}) \end{aligned}$$

(33)

satisfies $Gf_\varepsilon \ge 0$ if and only if

$$\begin{aligned} Gh-H_\varepsilon h\ge 0, \end{aligned}$$

(34)

where

$$\begin{aligned} H_\varepsilon h(x):=\sum _{y\in \mathcal{S}}G(x,y)\phi _\varepsilon \big (h(y)-h(x)\big ) \quad \text{ with } \ \phi _\varepsilon (z):=\varepsilon ^{-1}(e^{-\varepsilon z}-1+\varepsilon z).\nonumber \\ \end{aligned}$$

(35)

Proof

Let $g_\varepsilon (z):=\varepsilon ^{-1}(1-e^{-\varepsilon z})$ $(z\in {\mathbb {R}})$. Then, for any $z,z_0\in {\mathbb {R}}$,

$$\begin{aligned} g_\varepsilon (z)=g_\varepsilon (z_0)+\big \{(z-z_0)-\phi _\varepsilon (z-z_0)\big \}e^{-\varepsilon z_0}. \end{aligned}$$

(36)

It follows that

$$\begin{aligned} \displaystyle Gf_\varepsilon (x)= & {} \displaystyle \sum _{y\in \mathcal{S}}G(x,y) \big \{g_\varepsilon \big (h(y)\big )-g_\varepsilon \big (h(x)\big )\big \}\nonumber \\= & {} \displaystyle e^{-\varepsilon h(x)}\sum _{y\in \mathcal{S}}G(x,y) \big \{\big (h(y)-h(x)\big )-\phi _\varepsilon \big (h(y)-h(x)\big )\big \}, \end{aligned}$$

(37)

which is nonnegative if and only if (34) holds.