Fokker-Planck equation from <PERSON> equation in stochastic inflation
I'm reading this paper by <PERSON> and <PERSON> where they give the coarse-grained equation of motion, $$ \dot{\bar{\phi}}({\bf x},t ) = -\frac{1}{3H}V'(\bar{\phi}) + f({\bf x},t) $$ where $f({\bf x},t)$ is given by $$ f({\bf x},t) = \epsilon a(t)H^2 \int\frac{d^3 k}{(2\pi)^{3/2}}\delta(k - \epsilon a(t)H) \left[ a_{\bf k}\phi_{\bf k}(t) e^{-i {\bf k}\cdot {\bf x}} + a_{\bf k}^\dagger\phi_{\bf k}^*(t) e^{i {\bf k}\cdot {\bf x}} \right]. $$ This equation can be considered to be a <PERSON> equation for the stochastic quantity $\bar{\phi}$ with a stochastic noise term $f({\bf x},t)$ with correlation properties given by $$ \left\langle f({\bf x}_1, t_1) f({\bf x}_2, t_2) \right\rangle = \frac{H^3}{4\pi^2}\delta(t_1 - t_2) j_0(\epsilon a(t)H|{\bf x}_1 - {\bf x}_2|), \qquad j_0(z) = \frac{\sin z}{z}. $$ The authors then write down the Fokker-Planck equation corresponding to this <PERSON> equation, $$ \frac{\partial \rho[\bar{\phi}({\bf x},t)]}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\bar{\phi}} \left{ V'[\bar{\phi}({\bf x},t)]\rho[\bar{\phi}({\bf x},t)] \right} + \frac{H^3}{8\pi^2} \frac{\partial^2 \rho[\bar{\phi}({\bf x},t)]}{\partial \bar{\phi}^2}. 955
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Question: How do you derive this equation?
I've attempted to derive this using the Kramer-Moyel expansion. For a <PERSON> equation of the form, $$ \dot{\xi}(t) = h(\xi, t) + g(\xi, t)\Gamma(t) $$ where, $$ \left\langle \Gamma(t) \right\rangle = 0, \qquad \left\langle \Gamma(t)\Gamma(t') \right\rangle = 2\delta(t - t') $$
The corresponding <PERSON>-Planck equation is given by $$ \frac{\partial W(\xi,t)}{\partial t} = -\frac{\partial}{\partial \xi}[D^{(1)}(\xi,t )W(\xi, t)] + \frac{\partial^2}{\partial \xi^2}[D^{(2)}(\xi,t )W(\xi, t)] $$ where $D^{(i)}$ are the Kramer-Moyel expansion coefficients (see <PERSON>, The Fokker-Planck Equation) given by, $$ D^{(1)}(\xi,t) = h(\xi,t) + \frac{\partial g(\xi,t)}{\partial \xi} g(\xi,t), \qquad D^{(2)}(\xi,t) = g^2(\xi,t) $$
However, this result doesn't seem to work because in the problem of interest the stochastic force is dependent on both ${\bf x}$ and $t$. Additionally, the correlation $\left\langle f({\bf x}_1, t_1) f({\bf x}_2, t_2) \right\rangle$ is far more complicated in this case.
Is there a more general formalism that could help with more general force $f({\bf x},t)$?
Any help or resources about this particular derivation would be greatly appreciated. Thanks in advance! 955