In Lofstedt R and Coppersmith S N 1994 Phys. Rev. E 49 4821, in Appendix A, I came across a quite peculiar way to rewrite an integral, which I do not understand.
We start from the following: $$n_+(t) = \exp\left(-\int_{t_0}^t W(t') \rm{d}t' \right)n_+(t_0) + \exp\left(-\int_{t_0}^t W(t') \rm{d}t' \right) \int_{t_0}^t W_-(t') \exp\left( \int_{t_0}^{t'} W(t'')\rm{d}t'' \right)\rm{d}t' \tag{1}\label{eq1}$$ where $W_-(t)$ and $W_+(t)$ are periodic functions with period $t_s$, and $W(t) = W_-(t) + W_+(t)$ (suppose that everything is differentiable and defined everywhere on the real line).
Then, let $\delta W(t) = W(t) - \langle W \rangle$, where $\langle W \rangle$ is the average value of $W(t)$. Due to the periodicity of $W_-(t)$, we have, for any arbitrary reference point $t^*$:
$$\int_{t^* - 2 t_s}^{t^* - t_s} \rm{d}t' W_-(t') e^{\langle W \rangle t'} \exp\left(-\int_{t'}^t \delta W(t'') \rm{d}t'' \right) = e^{-\langle W \rangle t_s} \int_{t^* - t_s}^{t^*} \rm{d}t' W_-(t') e^{\langle W \rangle t'} \exp\left(-\int_{t'}^t \delta W(t'') \rm{d}t''\right ) \tag{2}\label{eq2} $$
Then, the paper's author states that $n_+(t)$ can be recast to: $$n_+(t) = e^{-\langle W \rangle(t-t_0)} h(t_0,t) n_+(t_0) - \frac{e^{-\langle W \rangle (t-t_0)}}{1 - e^{-\langle W \rangle t_s}} \int_0^{t_s} \rm{d}t_2 W_-(t_0-t_2) e^{-\langle W \rangle t_2} h(t_0-t_2,t) + \frac{1}{1 - e^{-\langle W \rangle t_s}} \int_0^{t_s} \rm{d}t_2 W_-(t-t_2) e^{-\langle W \rangle t_2} h(t-t_2,t) \tag{3} \label{eq3}$$ where the function $h$ is defined as follows: $$h(t_1,t_2) = \exp\left(-\int_{t_1}^{t_2} \delta W(t^*) \rm{d}t^* \right) $$
After many tries and several computations, I've been able to reconstruct \eqref{eq3} in the limit $t \to +\infty$. However, the general form of \eqref{eq3} for any $t$ still eludes me. Do you have any tips on how to use \eqref{eq1} and \eqref{eq2} to write \eqref{eq3}?