Consider a $1$ dimensional Brownian motion of a particle starting at $0$. Then, we know that the probability that the particle reaches the point $x$ at a time $\geq t_0$ is given by $$\mathbb{P}(\sup_{0 < t < t_0} B(t) > x) = 2\mathbb{P}(B(t_0) > x).$$
Also, $B(t_0)$ is a random variable with a normal distribution with mean $0$ and standard deviation $\sqrt{t_0}$. I have just started learning this stuff, and I am wondering if there is any analogous statement in higher dimensions, let's say, in $2$ dimensions. Do we have a mathematical treatment for the probability of a particle starting at $0$ reaching, let's say, the quarter circle $\{(x, y) \in \mathbb{R}^2 : x, y \geq 0\}$ within time $t_0$? Thanks!
Edit: Just cross-posted on MO, please answer on that site. Thanks!