Given a 3-dimensional Brownian motion $B_t$, we know that it is transient. But how can we show that if it starts outside a straight line, it will remain outside forever with probability $1$ ? Any ideas?
2026-03-30 12:18:11.1774873091
Hitting time of Brownian Motion on a line
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The orthogonal projection of $B_t$ on a plane normal to the line is $2$-dimensional Brownian motion, which with probability $1$ will never hit the origin.