Can anyone tell me where I am going wrong here? (I am leaving out any random fluctuation forcings, because I don't think they are relevant to my problem.)
1: $\displaystyle \frac{dv(t)}{dt}=-\eta v(t)$
2: $\displaystyle \frac{dv(t+\tau)}{dt}\frac{dv(t)}{dt}=\eta^2 v(t+\tau)v(t)$
3: $\displaystyle \frac{d}{d\tau}\left(v(t+\tau)\frac{dv(t)}{dt}\right)=\eta^2 v(t+\tau)v(t)$
4: $\displaystyle \frac{d}{d\tau}\left(\frac{d}{dt}(v(t+\tau)v(t))-\frac{dv(t+\tau)}{dt}v(t)\right)=\eta^2 v(t+\tau)v(t)$
5: $\displaystyle -\frac{d^2}{d\tau^2}\left(v(t+\tau)v(t)\right)=\eta^2 v(t+\tau)v(t)$
6: $\displaystyle \frac{d^2}{d\tau^2}\langle v(t+\tau)v(t)\rangle =-\eta^2 \langle v(t+\tau)v(t)\rangle$
That minus sign on the RHS can't be right, but I don't see where I am going wrong. Thanks.