Define the first hitting time $\tau^x_A:=\inf\{t>0, B^x\in A\}$, B is a standard Brownian motion, my question is: if $P(\tau_A^x<\infty)>0$, then $P(\tau_A^x<\infty)=1$? For 1, 2 dimensional Brownian motion, it is true. But what about dimension $\geq 3$?
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