The Coupled Branching Process in Random Environment

Abstract

We consider a Markov process $(\eta^\mu_t)_{t \in \mathbb{R}^+}$ on $(\mathbb{N}^S (S = \mathbb{Z}^d)$ with initial distribution $\mu$ and the following time evolution: At rate $b\sum_y q(y, x)\eta(y)$ a particle is born at site $x$; at rate $\tilde{d}\eta(x)$ a particle dies at site $x$. All particles perform independent from each other continuous time random walk with kernel $p(x, y)$ and rate $m$. All particles at a site $x$ die at rate $D(x)$. Here $D(x)$ are random variables taking the values $D_1, D_2 (D_2 \geq D_1 \geq 0)$. We assume $\{D(x)\}_{x \in S}$ to be stationary and ergodic. This paper studies the features of the model for $p(x, y), q(x, y)$ symmetric. We calculate the exponential growth rate $\lambda$ of $\tilde{E}(\eta^\mu_t(x))$ (with $\tilde{E}$ denoting conditional expectation with respect to the environment) and show that $\lambda$ is nonrandom and strictly bigger than $b - \tilde{d} - E(D(x))$, if $D_2 > D_1$. We have $\lambda = b - \tilde{d} - D_1$. Introduce the process $(\hat{\eta}^\mu_t)_{t\in \mathbb{R}^+}$ by setting $\hat{\eta}^\mu_t(x) = (\tilde{E}(\eta^\mu_t(x)))^{-1} \eta^\mu_t(x)$. A critical phenomenon with respect to the parameter $p := D_1(\tilde{d} + ED(x))^{-1}$ occurs in the sense that for $p > p^{(2)}$ the quantity $\tilde{E}(\hat{\eta}^\mu_t(x))^2$ grows exponentially fast, while for $p \leq p^{(2)}, \lambda > 0$ the exponential growth rate of $\tilde{E}(\hat{\eta}^\mu_t(x))^2$ is 0. $p^{(2)}$ is the same as for a system with $D(x) \equiv D_1$ and can be calculated explicitly.