I'm kind of not 100% sure about this: Let $B$ be a Brownian Motion defined on some Probabitlity space $(\Omega, \mathscr{F}_t,\mathbb{P)}$ and $\phi^{-1}:\mathbb{R}\rightarrow\mathbb{R}$ increasing continuous function. Is it true that for any $\varepsilon>0$ $ (\varepsilon\in \mathbb{R})$ and all $t\in \mathbb{R}$: $$\varepsilon B(\varepsilon^{-2}\phi^{-1}(t))\stackrel{d}{=}B(\phi^{-1}(t))$$
Should be true but i might be wrong.
