Hitting distribution to a quadrant of two-dimensional random walk
Let \((S(n))_{n\in\mathbb N_0}\) be a simple random walk on \(\mathbb Z^2\), starting from \(\xi\in\mathbb Z^2\) under \(P_\xi\). Denote by \(\tau_L=\inf\{n\in\mathbb N: S(n)\in L\}\) the hitting time of the third quadrant, \(L=(-\infty,0]^2\), and by \(H_L(\xi,\eta)=P_\xi(S(\tau_L)=\eta)\) the hitting distribution on the boundary of \(L\) when started at \(\xi\notin L\). The Brownian analogues are defined accordingly: For a standard Brownian motion \((W(t))_{t\geq 0}\) on \(\mathbb R^2\), let \(T_L\) be the hitting time of \(L\), and let \(h_L(\xi,\cdot)\) denote the density of the hitting distribution of \(L\), when the motion starts at \(\xi\in\mathbb R^2\setminus L\). The density \(h_L(\xi,\cdot)\) may be explicitly calculated in terms of certain Bessel functions. It is the aim of the paper to derive estimates for the hitting distribution of the simple random walk in terms of the one for the Brownian motion. More precisely, the main result states that, for some absolute constant \(C_0>0\), we have the estimate \[ |H_L(\xi,(-l,0))-h_L(\xi,(-l,0))|\leq C_0[|\xi+(l,0)|^{-3}+|\xi|^{-2/3} l^{-5/3}] \] for any \(\xi\in\mathbb Z^2\setminus L\) and any \(l\in\mathbb N\). Furthermore, it is shown that the powers \(2/3\) and \(5/3\) cannot be improved. Moreover, the existence and positivity of the limits \(\lim_{l\to\infty}l^{5/3}H_L(k,(-l,0))\) and \(\lim_{k\to\infty}k^{2/3}H_L(k,(-l,0))\) for any fixed \(k\in\mathbb N\) resp.\ \(l\in\mathbb N\) are shown. The main tools are estimates for the Green's function of the walk, the reflection principle and an extensive analysis.
two-dimensional standard Brownian motion
scientific article; zbMATH DE number 1496083