Integrate the Langevin equation (1) of Problem 15.4 directly over the amall time interval $\tau$ to find $\Delta v=v(\tau)-v_{\mathrm{n}}$
(a) Use this result to express $\overline{\Delta v}$ and $\overline{(\Delta v)^2}$ in terms of $G$ and $\overline{G^2}$. Show that these moments are proportional to r and find their explicit values by using the results of Problems 15.7 and 15.8 .
(b) Express $\overline{(\Delta v)^3}$ and $\overline{(\Delta v)^2}$ in terms of moments of $G$; show that these quantities are proportional to $\tau^2$.
(c) Find an explicit expression for $\overline{(\Delta y)^3}$.