Help on calculation in <PERSON>, "Nonequilibrium Statistical Mechanics"
I'm trying to verify a calculation from Zwanzig's Nonequilibrium Statistical Mechanics text from chapter two, deriving the Fokker-Planck Equation from a Langevin Equation, specifically equation (2.41).
Define the <PERSON> equation, \begin{equation}\dot{\boldsymbol{a}}=\boldsymbol{v}(\boldsymbol{a})+\boldsymbol{F}(t)\end{equation} where $\boldsymbol{v}$ is a general function of $\boldsymbol{a}$, and $\boldsymbol{F}$ is a Gaussian, white noise process: $$\langle\boldsymbol{F}(t)\rangle_{t}=\boldsymbol{0}\ \langle\boldsymbol{F}(t)\boldsymbol{F}(s)\rangle_{t}=2\boldsymbol{\mathbf{B}}\delta(t-s)$$ Seeking a pdf $f(\boldsymbol{a},t)$ for $\boldsymbol{a}$, we invoke a conservation law over the state space of $\boldsymbol{a}$: $$\frac{\partial f}{\partial t}=-\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f\,\frac{\partial\boldsymbol{a}}{\partial t}\right)$$ Using the <PERSON> equation for $\boldsymbol{a}$, we have $$\frac{\partial f}{\partial t}=-\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},t)\,\boldsymbol{v}(\boldsymbol{a})+f(\boldsymbol{a},t)\boldsymbol{F}(t)\right)$$ Define an operator $K$ by $$K\,u=\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(\boldsymbol{v}(\boldsymbol{a})\,u\right)$$ We can rewrite the conservation law, $$\frac{\partial f}{\partial t}=-K\,f-\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},t)\boldsymbol{F}(t)\right) \tag{1}$$
The noiseless conservation law then admits an operator solution, $$f(\boldsymbol{a},t)=e^{-tK}f(\boldsymbol{a},0)$$ With noise, we can write a solution $$f(\boldsymbol{a},t)=e^{-tK}f(\boldsymbol{a},0)-\int_0^te^{-(t-s)K}\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},s)\boldsymbol{F}(s)\right)\,ds \tag{2}$$
What I'm stuck on is how substituting equation (2) into the right-hand side of equation (1) is supposed to beget, $$\frac{\partial f}{\partial t}=-K\,f -\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},0)F(t)\right) +\frac{\partial}{\partial\boldsymbol{a}}\cdot\left[F(t)\int_0^te^{-(t-s)K}\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},s)\boldsymbol{F}(s)\right)\,ds\right] \tag{3}$$
Specifically, I can't get the middle term, $\frac{\partial}{\partial\boldsymbol{a}}\cdot\left(f(\boldsymbol{a},0)F(t)\right)$. 955