I am trying to calculate the following probability:
$$P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq T\}$$ where $a, T > 0$.
By scaling property, for any $b > 0$, this probability is equal to $$P\{bW_{\frac{t}{b^2}} \leq a\sqrt{t} \quad \forall{t} \leq T\}$$ Defining $u =\frac{t}{b^2}$, $$P\{W_u \leq a\sqrt{u} \quad \forall{u} \leq \frac{T}{b^2}\}$$
I choose $b = \sqrt{T}$ to get $$P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq T\} = P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq 1\}$$
By time reversal,
$$\begin{eqnarray} P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq 1\} &=& P\{tW_{\frac{1}{t}} \leq a\sqrt{t} \quad \forall{t} \leq 1\} \nonumber \\ &=& P\{W_t \leq a\sqrt{t} \quad \forall{t} \geq 1\} \nonumber \\ \end{eqnarray}$$
But this implies
$$P\{W_t \leq a\sqrt{t} \quad \forall{t} \geq 0\} = 2P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq 1\}$$ Hence $$P\{W_t \leq a\sqrt{t} \quad \forall{t} \leq 1\} = 0$$
I am looking for alternative approaches to this problem (e.g. slick one-liners, convoluted derivations using martingale theory)
By law of iterated logarithm we have a.s. $\limsup_{t\to 0^+}\frac{W_t}{(2t \ln|\ln t|)^{1/2}}=1$ (ref.: Schilling, Corollary 12.2.) so $$\limsup_{t\to 0^+}\frac{W_t}{t^{1/2}}=\limsup_{t\to 0^+}\frac{W_t}{(2t \ln|\ln t|)^{1/2}}(2 \ln|\ln t|)^{1/2}=\infty$$ a.s. So the probability of the event is zero.