Let $W$ be a standard Brownian Motion indexed by $\mathbb{R}$, i.e., $W\restriction_{\mathopen{[}0,\infty\mathclose{[}}$ is a standard Brownian Motion, and $W\restriction_{\mathopen{]}-\infty,0\mathclose{]}}$ is another standard Brownian Motion independent of $W\restriction_{\mathopen{[}0,\infty\mathclose{[}}$, except that it is indexed in the reverse order. Let $f$ be a function in the Schwartz space
\begin{equation}\mathscr{S}(\mathbb{R}) = \left\{f\in\mathscr{C}^\infty(\mathbb{R})\colon \max_{j=0,\dots,n}\sup_{u\in\mathbb{R}}\left\lvert(1+u^2)^n f^{(k)(u)}\right\rvert<\infty,\forall n\in\mathbb{N}\cup\{0\}\right\}.\end{equation}
I want to show that the integral $\int_{\mathbb{R}}W(t)f^\prime(t)~\!\mathrm{d}t$ is defined, i.e.,
$$\int_{\mathbb{R}}\lvert W(t)f^\prime(t)\rvert~\!\mathrm{d}t<\infty.$$
I have no idea where to start on this one. My thinking is that one of the Brownian growth results, like the Law of the Iterated Logarithm, might help, or perhaps something like an approximation by polynomials of the Brownian paths (motivated by Stone–Weierstraß) so that the fact that polynomials are tempered distributions (in the $\mathscr{L}^2(\mathbb{R})$-dual of $\mathscr{S}(\mathbb{R})$) comes into play. But, I've made no progress with either idea. Any help will be appreciated.
Law of the Iterated Logarithm is plenty. It implies, in particular, that $$\sup_{t \in (-\infty,+\infty)} \frac{|B(t)|}{\sqrt{ 2 |t| \log \log |t|}} < \infty$$ almost surely. Or in other words, there is a random variable $C$, with $C < \infty$ almost surely, such that $|B(t)| \le C \sqrt{ 2 |t| \log \log |t|}$.
Now because $f$ is a Schwartz function, we can say, for instance, that there is a constant $K$ with $|f'(t)| \le K/({(1+t^2)^{42}})$ (overkill). Hence $$\int_{\mathbb{R}} |B(t) f'(t)|\,dt \le CK \int_{\mathbb{R}} \frac{\sqrt{ 2 |t| \log \log |t|}}{(1+t^2)^{42}}\,dt$$ which is obviously finite.