Suppose $W_t= (X_t,Y_t)$ is a $2$d standard Brownian motion starting at $(-1,0)$. How do I show that there is a positive probability that $W_t$ moves from $(-1,0)$, to a neighborhood of $(1,0)$, say the ball $B((1,0),1/8)$, without touching the rectangle enclosed by $x=-2, y=1,y=-1,x=2$.
I was able to show that $P(Y_s\in (-1/8,1/8),\forall s\leq t)>0$ for any $t$. and that $\exists t, P(\text{before time } t, X_s \text{ hit the interval } [7/8,9/8] \text{ before hitting } -2)>0$ and the conclusion should follow.
But I am looking for a simple or elegant way in doing this as the question seems to be really basic and maybe obvious to some of you.