= Infinite occupation time of one-dimensional Brownian motion
A one-dimensional <Brownian motion> spends an infinite total time in every nonempty open interval <almost surely>. To prove this for $(-1,1)$ after first hitting zero, alternate exits from that interval with returns to zero. By <recurrence of one-dimensional Brownian motion> all these <stopping times> are finite, and the <Strong Markov property> makes the successive exit durations <independent and identically distributed random variables> that are strictly positive. Some fixed positive duration is exceeded with positive probability, so the <Borel-Cantelli lemma> makes the sum of the durations infinite. Translate and scale for other intervals. In particular, if a nonnegative function is bounded below by a positive constant on such an interval, its infinite-horizon <Brownian occupation time> integral diverges. Recurrent visits alone, without this duration argument, would not prove divergence of an integral.
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