ИСТИНА

We consider a symmetric, homogeneous (in space and in time), irreducible random walk $S=\{S(t),t\geq0\}$ on $\mathbb{Z}^{d}$, $d\in\mathbb{N}$, having a finite variance of jumps. Denote by $\tau$ the time of the first exit of the process $S$ from the starting point, that is, $\tau:=\inf\{t\geq0: S(t)\neq S(0)\}$. For $y,z\in\mathbb{Z}^{d}$, $y\neq z$, we introduce the notion of \emph{hitting time of a point $y$ with a taboo state $z$} by way of $\tau_{y,z}:=\inf\{t\geq\tau: S(t)=y, S(u)\neq z, \tau\leq u\leq t\}$ where as usual $\inf\{t\in\varnothing\}=\infty$. Loosely speaking, for trajectories of the random walk, not passing the taboo state $z$ before the first hitting of the state $y$, the (extended) random variable $\tau_{y,z}$ equals the time of the point $y$ hitting. For the rest trajectories, one has $\tau_{y,z}=\infty$. Set $H_{x,y,z}(t):={\sf P}(\tau_{y,z}\leq t|S(0)=x)$, $x,y,z\in\mathbb{Z}^{d}$, $y\neq z$, $t\geq0$. Our main results concern the asymptotic properties of the (improper) cumulative distribution functions $H_{x,y,z}(t)$, as $t\to\infty$. Firstly, we find the explicit formula for the limit value $H_{x,y,z}(\infty)=\lim\nolimits_{t\to\infty}{H_{x,y,z}(t)}$. Secondly, the asymptotic behavior of $H_{x,y,z}(\infty)-H_{x,y,z}(t)$, as $t\to\infty$, is established for arbitrary $x,y,z\in\mathbb{Z}^{d}$, $y\neq z$. In particular, we show that for the random walk on $\mathbb{Z}^{d}$, except for a simple (nearest-neighbor) random walk on $\mathbb{Z}$, $H_{x,y,z}(\infty)$ belongs to the interval $(0,1)$. In contrast, for a simple random walk on $\mathbb{Z}$, $H_{x,y,z}(\infty)$ can take values $0$, $1$ or belong to $(0,1)$ depending on the relative positions of $x$, $y$ and $z$. Moreover, for the random walk on $\mathbb{Z}^{d}$, except for a simple random walk on $\mathbb{Z}$, the order of decrease of $H_{x,y,z}(\infty)-H_{x,y,z}(t)$, as $t\to\infty$, is determined by dimension $d$ only. However, for a simple random walk on $\mathbb{Z}$, this is specified by the mutual location of $x,$ $y$ and $z$. Note that establishing these asymptotic properties is pertinent to our recent study of branching random walk on $\mathbb{Z}^{d}$ with a single source of branching.