Consider a noise-driven drifting system given by the Langevin Eq:
$$\dot{v}+\gamma v =\xi$$ Where $v$ denotes velocity with $\gamma$ being drift coefficient and $\xi$ a stochastic-noisy drive, but unlike Orstein-Uhlenbeck Process, $\xi 's$ are not $uncorrelated$ rather they themselves are $\frac{1}{f}$ noise with power-law autocorrelation function [recalling from Wiener-Khinchin theorem that the autocorrelation function is fourier transform of the power spectral density where $\frac{1}{f}$ noise has Power Spectral density $S(f)\propto \frac{1}{f^{\alpha}}$ with $1 \leq \alpha \leq 2$], say (assuming stationarity):
$$\langle \xi(t) \ \xi(0) \rangle = \tilde{\Gamma}|t|^{-\theta} \ \ ; \ \ \ 0 \leq \theta \leq 1$$
Therefore the velocity profile of $v(t)$ is as:
$$v(t)= \Big[ v_0 + \int^{t}_{0}e^{\gamma t'} \xi(t')dt' \Big] e^{-\gamma t} $$ where $\langle \xi(t) \rangle = 0$ will be considered, thus the average $\langle v(t) \rangle$ is same as Orstein-Uhlenbeck case,
BUT, what about the case of average energy, or second moment $\langle v^2(t) \rangle$, i.e.:
$$\langle v^2(t) \rangle = v_0^2 e^{-2 \gamma t} + e^{- 2 \gamma t } \int^{t}_{0} dt_1 \int^{t}_{0} dt_2 e^{\gamma{(t_1 + t_2)}} \langle \xi(t_1) \xi(t_2) \rangle $$
This expression is well treatable with delta correlated gaussian white noise, but it's nontrivial here.
How do I get passed with this integral ?
$$\int^{t}_{0} dt_1 \int^{t}_{0} dt_2 e^{\gamma{(t_1 + t_2)}}|t_1 - t_2|^{-\theta} $$
I am stuck here. Kindly tell me how to handle such long-range-correlated-noisy setup.