= Solution
Let
$$
t_{\min} =\min(t_2,t_3),
\qquad
t_{\max} =\max(t_2,t_3).
$$
Evolution from the earlier time to the later time multiplies the earlier field by the deterministic decay factor, plus fresh noise independent of that field. The unequal-time <correlation function> is therefore
$$
\begin{aligned}
\langle p_{\mathbf q,i}(t_2)p_{-\mathbf q,j}(t_3)\rangle
=\delta_{ij}e^{-r(q)|t_3-t_2|}
\bigg[
&\frac{k_BT}{a_I+\kappa q^2}
e^{-2r(q)(t_{\min}-t_1)}
\\
&+\frac{k_BT}{a_F+\kappa q^2}
\left(1-e^{-2r(q)(t_{\min}-t_1)}\right)
\bigg].
\end{aligned}
$$
Equivalently,
$$
\begin{aligned}
\langle p_{\mathbf q,i}(t_2)p_{-\mathbf q,j}(t_3)\rangle
=\delta_{ij}\bigg[
&\frac{k_BT}{a_F+\kappa q^2}e^{-r(q)|t_3-t_2|}
\\
&+\left(
\frac{k_BT}{a_I+\kappa q^2}
-\frac{k_BT}{a_F+\kappa q^2}
\right)
e^{-r(q)(t_2+t_3-2t_1)}
\bigg].
\end{aligned}
$$
When both observation times are many relaxation times after the quench, the second term vanishes and the stationary <Ornstein-Uhlenbeck process> covariance remains:
$$
\langle p_{\mathbf q,i}(t_2)p_{-\mathbf q,j}(t_3)\rangle
\longrightarrow
\delta_{ij}\frac{k_BT}{a_F+\kappa q^2}
e^{-r(q)|t_3-t_2|}.
$$
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