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Let $p\in(0,1)$. The corresponding quantile function of $X\sim N(\mu,\sigma^2)$ is given by $$F_X^{-1}(p)=\mu+\sigma\Phi^{-1}(p)=\mu+\sqrt{2}\sigma\mathrm{erf}^{-1}(2p-1),$$ where $\Phi^{-1}$ is the inverse of the cumulative distribution function of a standard normally distributed random variable and $\mathrm{erf}^{-1}$ is the inverted error function. Thus, ...
In some books, what you want to prove is just part of the definition of the Brownian motion. In others, as part of the definition of the B.M., they give the following condition: $$\text { for } 0 \leq s < t < \infty, W_t - W_s \ \text{is independent of } \mathscr{F}_s \tag*{(*)}$$ So, I'm assuming that given (*) you want to prove that the random ...