Let $B_t$ be Brownian motion for $t\ge0$ so that $B_t-B_s \sim N(0,t-s)$. There are a few formulae out there that show how the $p$-th moment of $E[(B_t-B_s)^p]$ is calculated.
My question is: how do they scale with time, i.e. how does $E[(B_t-B_s)^p]$ relate to $E[B_{(t-s)/2+s}-B_s)^p]$ and so on? An easier way of capturing this would be how does $E[B_{t}^p]$ relate to $E[B_{t/2}^p]$ since we can just look at the first increment. It would be nice to see an answer that only uses the scaling and shifting properties of Brownian motion without using the explicit clunky expressions for the $p$-the moments but any answer would help.