Nakajima-Zwanzig-Gleichung

Nakajima Zwanzig Equation $\begin{array}{cc}& d_t\chi =L\chi \\ & \chi =𝒫\chi +𝒬\chi \\ & d_t\left(\begin{array}{c}𝒫\\ 𝒬\end{array}\right)\chi =\left(\begin{array}{c}𝒫\\ 𝒬\end{array}\right)L\left(\begin{array}{c}𝒫\\ 𝒬\end{array}\right)\chi +\left(\begin{array}{c}𝒫\\ 𝒬\end{array}\right)L\left(\begin{array}{c}𝒬\\ 𝒫\end{array}\right)\chi \\ & \Rightarrow 𝒬\chi =e^{𝒬Lt}Q{\chi }_0+{\int }^{\prime }{e}^{𝒬Lt}𝒬L𝒫\chi (t-t^{\prime })\\ & \Rightarrow {\text{d}}_t𝒫\chi =𝒫L𝒫\chi +\underset{=0}{\underbrace{𝒫e^{𝒬Lt}Q{\chi }_0}}+𝒫L{\int }^{\prime }{e}^{𝒬Lt}𝒬L𝒫\chi (t-t^{\prime })\\ \end{array}$