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First prove the strong law for Brownian motion, $B_t/t\to 0$ with probability one. For each $\epsilon >0$, the Gaussian tail bound givesBy stationary increments and the Brownian reflection principle, followed by the same tail bound,Both bounds are summable in $n$. The First Borel-Cantelli lemma therefore implies that, eventually, both quantities inside these probability events are at most $\epsilon n$. For $n\le t\le n+1$ this gives $|B_t|/t\le 2\epsilon$. Intersect over a sequence of positive rational $\epsilon$ tending to zero to obtain the asserted continuous-time limit.

On this one almost sure event,so $B_t+at\to +\infty$ for $a>0$ and to $-\infty$ for $a<0$. For each real $x$, the path is eventually strictly on the corresponding side of $x$. HenceThe same event works simultaneously for all levels $x$, giving the requested transience of Brownian motion with drift.

Use the Kolmogorov continuity theorem in its one-parameter form: if on a compact interval a process satisfies $\mathbb{E}|X_t-X_s{|}^p\le C|t-s{|}^{1+\beta }$ for some $p,\beta >0$, it has a modification whose paths are Hölder continuous of every order $\gamma <\beta /p$ on that interval.

For Brownian motion, normal increments give, for every $p>0$,Every such normal moment is finite. Taking $p>2$ yields $\beta =p/2-1$, hence any order below $1/2-1/p$. For a prescribed $0<\alpha <1/2$, choose $p$ with $1/2-1/p>\alpha$.

The continuous modification and the given continuous Brownian motion agree at all rational times on one almost sure event; continuity makes them agree everywhere on the interval. To obtain all exponents and all compact intervals simultaneously, apply the theorem to integer intervals $[0,N]$ and a countable sequence of positive exponents increasing to $1/2$, then intersect these almost sure events. A bound at exponent $\gamma >\alpha$ implies a bound at $\alpha$ on a compact interval. Consequently the Brownian Hölder regularity conclusion iswith finite random constants on one common event of probability one. This uses the usual positive-exponent meaning of Hölder continuity.