Consider $(S_n)_n$ the simple random walk in two dimensions starting from $x \in B_n:=\{y: ||y||_2 \le n\}$ for a given $n \ge 1$. Denote its law by $\mathbb P_x$.
Let $\tau_{n}:= \inf\{n \ge 1: S_n \in \{0\}\cup \partial B_n\}$ where $\partial B_n$ denotes the exterior border of $B_n$ in the sense of graphs.
My main question is: What is $\mathbb P_x(S_{\tau_n}=0)$?
Is it possible to get a closed expression for this? If not, choosing another norm (instead of $||\cdot||_2$) help? If the answer is still negative, could we get an asymptotic expression ($||x||_2$ and $n$)?
This is an analogous to the well-known Gambler's ruin problem. However, here it is not clear how to choose the right martingale to be able to derive the quantities we need.
The asymptotic result is presented in this paper. (See Section 2 on page 7). Specifically, if $\tau_R:=\inf\{n\ge 1: S_n\in\partial B(0,R)\}$ and $\tau_0:=\inf\{n\ge 1: S_n=0\}$, for $x\in B(0,R)\setminus \{0\}$ and $R\ge 1$, the probability that $S_n$ started at $x$ hits $\partial B(0,R)$ before the origin is given by $$ \mathsf{P}(\tau_R<\tau_0)=\frac{a(x)}{a(R)+O(R^{-1})}, $$ as $R\to\infty$, where $$ a(r)=\frac{2}{\pi}\ln r+\frac{2\gamma+3\ln 2}{\pi}. $$