Branching random walk in random environment with random absorption wall

We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^\alpha-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,\alpha=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,\alpha=\frac{1}{3}$ ,the survival probability is zero almost surely. Moreover, if we denote $Z_n$ is the total populations in $n$-th generation in the new system (with barrier),under some conditions, we show $\ln\mathbb{P}_{\mathcal{L}}(Z_n>0)/n^{1/3}$ will converges to a negative constant almost surely if $\alpha\in[0,\frac{1}{3})$.