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We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment $\{\mathcal{L}_n\}$ indexed by time $n,$ i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation $n$ ($n\in\mathbb{N}$), the barrier is set as $\chi_n+\varepsilon n,$ where $\{\chi_n\}$ is a random walk determined by the random environment. Lv \& Hong (2024) showed that for almost every $\mathcal{L}:=\{\mathcal{L}_n\},$ the quenched survival probability (denoted by $\varrho_{\mathcal{L}}(\varepsilon)$) of the particles system will be 0 (resp., positive) when $\varepsilon\leq 0$ (resp., $\varepsilon>0$). In the present paper, we prove that $\sqrt{\varepsilon}\log\varrho_\mathcal{L}(\varepsilon)$ will converge in Probability/ almost surely/ in $L^p$ to an explicit negative constant (depending on the environment) as $\varepsilon\downarrow 0$ under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.

Consider a supercritical branching random walk in a time-inhomogeneous random environment. We impose a selection (called barrier) on survival in the following way. The position of the barrier may depend on the generation and the environment. %to indexed by time $n,$ i.e.,$\{\L_n\}$with a random absorption barrier In each generation, only the individuals born below the barrier can survive and reproduce. When the barrier causes the extinction of the system, we give the extinction rate in the sense of $L^p~(p\geq 1)$. Moreover, we show the $L^p$ convergence of the small deviation probability for a random walk with random environment in time., Comment: 26 Pages

Consider the invariance principle for a random walk with random environment (denoted by $\mu$) in time on $\bfR$ in a weak quenched sense. We show that a sequence of the random probability measures on $\bfR$ generated by a bounded Lipschitz functional $f$ and $\mu$ will converge in distribution to another random probability measures, which is related to $f$ and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense., Comment: 13 pages, 0 figures

We consider a supercritical branching random walk in time-inhomogeneous random environment with a random absorption barrier, i.e.,in each generation, only the individuals born below the barrier can survive and reproduce. Assume that the random environment is i.i.d..The barrier is set as $\chi_n+an^{\alpha},$ where $a,\alpha$ are two constants and $\{\chi_n\}$ is a certain i.i.d. random walk determined by the random environment.We show that for almost surely given environment (i.e., a sequence of point processes which is a realization of the random environment), the time-inhomogeneous branching random walk under the given environment will become extinct (resp., survive with positive probability) if $\alpha<1/3$ or $\alpha=1/3, a1/3, a>0$ or $\alpha=1/3, a>a_c$), where $a_c$ is a positive constant determined by the random environment. The rates of extinction when $\alpha<\frac{1}{3}, a\geq0$ and $\alpha=1/3, a\in(0,a_c)$ are also obtained. These extend the main results in A\"{\i}d\'{e}kon $\&$ Jaffuel (2011) and Jaffuel (2012),to the random environment case. The influence caused by the random environment have been specified.

We consider the small deviation probability for random walk with time-inhomogeneous random environment. Compared with the result in Mogul'ski\u{\i} (1974) for the i.i.d. random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance., Comment: 21 pages, 0 figures

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 $a0)/n^{1/3}$ will converges to a negative constant almost surely if $\alpha\in[0,\frac{1}{3})$., Comment: 18 pages

We give the random environment version of Mogul'ski\v{\i} estimation in quenched sense.Assume that $\{\mu\}_{n\in\bfN}$ (called environment) is a sequence of i.i.d. random probability measures on $\bfR.$~ Let $\{X_n\}_{n\in\bfN}$ be a sequence of independent random variables, where $X_n$ has law $\mu_n.$ We set $S_n=\sum_{i=1}^{n}X_i.$ Under some integrability conditions, we show that on the log scale, for any power function $f$. the decay rate of $$\bfP_\mu(\forall_{0\leq i\leq n} S_{f(n)+i}\in[g(i/n)n^{\alpha},h(i/n)n^{\alpha}]|S_{f(n)}=x)$$ is $e^{-cn^{1-2\alpha}}$ almost surely as $n\rightarrow+\infty$, where $c>0,\alpha\in(0,\frac{1}{2}),$ $g,h\in\mathcal{C}[0,1]$ (the set of all continuous functions defined on $[0,1]),$ $g(s)

Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other one-dimensional Brownian motion independent of $\{B_s\}_{s\geq 0}$ and let $\bfP(\cdot|W)$ represent the conditional probability depending on the realization of $\{W_s\}_{s\geq 0}$. We show that $$-t^{-1}\ln\bfP^x(\forall_{s\in[0,t]}a+\beta W_s\leq B_s\leq b+\beta W_s|W)$$ converges to a finite positive constant $\gamma(\beta)(b-a)^{-2}$ almost surely and in $L^p~ (p\geq 1)$ if $a