 Solution

False. In fact, standard one-dimensional Brownian motion satisfies ${\mathrm{liminf}}_{t\to \infty }{B}_t=-\infty$ almost surely. For each integer $m\ge 1$, let ${\tau }_{-m}=\inf \{t:B_t=-m\}$. The Brownian reflection principle, applied to $-B$, giveswhere $\Phi$ is the standard normal distribution function. Thus each negative integer is reached almost surely. Intersecting these countably many probability-one events, all negative integer levels are reached on the same path. Their hitting times must become arbitrarily large, because a continuous path is bounded on every compact time interval. Hence the path cannot tend to positive infinity. Applying the same argument to positive levels also gives ${\mathrm{limsup}}_{t\to \infty }{B}_t=+\infty$ almost surely.