Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 )$, $(k_1 , k_2 ± 1)$ of p is selected with equal probability $\frac{1}{4}$ . The particle moves to that neighbor at time $t + 1$. Let $D$ be a region in the plane (a square or a half plane for example), and let $B$ denote its boundary. Let $p$ be a point of $D$, and let $b$ be a boundary point. We’ll denote by $P_p(b)$ the probability that a random walk starting at $p$ exits at $b$, i.e., that $b$ is the first boundary point that is reached.
I was wondering if someone could help me answer some questions if the region in the plane that we are considering is a rectangle.
1) What is the probability that a particle starting at $p$ never reaches the boundary? 2) What is the “exit time”, the expected time for a particle starting at $p$ to reach the boundary? 3) How does the exit time depend on $p$?
