In a PDF about Gaussian-Stochastic models for spectral diffusion, there is a simplification made by change of variable between equations 7.21 and 7.22 that I just can't seem to make.
Writing it here,
$$\tfrac{1}{2}\int^t_0 d\tau' \int^t_0 d\tau'' \langle d\omega(\tau' - \tau'') d\omega(0) \rangle ~~~~~~~~~~~~7.21$$
Substitute in $\tau = \tau' - \tau''$
$$ \int^t_0 d\tau (t - \tau) \langle d\omega(\tau) d\omega(0) \rangle ~~~~~~~~~~~~7.22$$
How did it happen? Please show me the intermediate steps.
My attempt
$$\tfrac{1}{2}\int^t_0 d\tau' \int^t_0 d\tau'' \langle d\omega(\tau' - \tau'') d\omega(0) \rangle $$ $$d\tau = d\tau' - d\tau''$$ $$\int^t_0 d\tau' \int^t_0 (d\tau' - d\tau) \langle \delta \omega( \tau )\delta \omega(0) \rangle $$ $$\int^t_0 d\tau' \int^t_0 d\tau' \langle \delta \omega ( \tau )\delta \omega(0) \rangle - \int^t_0 d\tau' \int^t_0 d\tau \langle \delta \omega( \tau )\delta \omega(0) \rangle $$ $$ t^2\langle \delta \omega ( \tau )\delta \omega (0) \rangle - \int^t_0 d\tau' \int^t_0 d\tau \langle \delta \omega ( \tau )\delta \omega (0) \rangle$$ It seems like I am making some obvious mistake, maybe with $d\tau = d\tau' - d\tau''$?