introduce an auxiliary field $h({\bm z}_1, t)$, that satisfies the following equation $$\begin{aligned}
 \partial_t h({\bm z}_1, t) + {\mathcal L}_{11} [h] = \xi({\bm z}_1, t),
\label{Eqh}\end{aligned}$$ where $\xi$ is a noise term with the following properties $$\begin{aligned}
& \langle \xi({\bm z}_1, t) \rangle = 0, 
\label{noiseaverage}\\
& \langle \xi({\bm z}_1, t) \xi({\bm z}_2, t')\rangle = 2 \lambda \frac{n}{4\pi} \delta(t-t') \delta({\bm x}_1 - {\bm x}_2 ) 
\nonumber \\
&\qquad\qquad \times \left[ \delta({\bm p}_1 - {\bm p}_2 ) - \frac{1}{4\pi} \right].
\label{noisevariance}\end{aligned}$$ Here, the angular brackets denote the average over the realisations of the noise $\xi$, and should not be confused with the ensemble averages that we denoted by bars in the equations above. Eq. allows us to factorise the Klimontovich correlation function as $$\begin{aligned}
G_K({\bm z}_1,{\bm z}_2, t) = \langle h({\bm z}_1, t) h({\bm z}_2, t)\rangle,\end{aligned}$$ which replaces the deterministic Eq. by a significantly simpler stochastic Eq. with a fictitious noise $\xi$ with properly chosen spectral properties. Remarkably, the non-equal time correlations of the phase space density can be expressed through the same auxiliary field $$\begin{aligned}
\overline{\delta \varphi({\bm z}',t') \delta \varphi({\bm z}'',t'')} = \langle h({\bm z}', t') h({\bm z}'', t'')\rangle,\end{aligned}$$ as implied by a seminal work of Klimontovich and Silin [@Silin1962]. 