Let $(\Omega,\mathcal A,\Bbb P)$ be a complete probability space. Let us consider a standard Brownian Motion $B=(B_t)_{t\ge0}$ on this space.
We know that its trajectories are $\alpha$-Holder continuous for every $\alpha<1/2$, and moreover that a.s. $B_0=0$.
Thus I was asking myself (even looking at the many realizations that can be found on the web): is it possible to control the behaviour of the trajectories near $0$?
I mean, I conjectured that $\forall \epsilon>0\;\;\exists\delta>0$ such that $$ t^{1/2+\epsilon}\le|B_t|\le t^{1/2-\epsilon}\;\;\forall t\in[0,\delta]. $$ This seems reasonable, but I don't know how to prove or disprove it.
EDIT: GENERALIZATION
If $B^H=(B_t^H)_{t\ge0}$ is a fractional Brownian Motion of Hurst parameter $0<H<1$, is it possible to prove that $$ t^{H+\epsilon}\le|B_t^H|\le t^{H-\epsilon}\;\;\forall t\in[0,\delta]? $$
You are looking for the law of the iterated logarithm for Brownian motion.
With probability one,
$$\limsup_{h\downarrow 0}{|B(h)|\over \sqrt{2h\log\log(1/h)}}=1.$$
This is Corollary 5.3 (top of page 121) in Brownian motion by Peter Mörters and Yuval Peres. You can download the book at yuvalperes.com/brbook.pdf