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I'm reading this paper by Starobinsky and Yokoyama 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 Langevin 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 Langevin equation, $$ \frac{\partial \rho[\bar{\phi}({\bf x},t)]}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\bar{\phi}} \left\{ V'[\bar{\phi}({\bf x},t)]\rho[\bar{\phi}({\bf x},t)] \right\} + \frac{H^3}{8\pi^2} \frac{\partial^2 \rho[\bar{\phi}({\bf x},t)]}{\partial \bar{\phi}^2}. $$

Question: How do you derive this equation?

I've attempted to derive this using the Kramer-Moyel expansion. For a Langevin 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 Fokker-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 Risken, 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!

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For the purpose of deriving the Fokker-Planck equation, we only need correlation between two different times $t_1$ and $t_2$ for the stochastic force, not for two different spatial locations. In other words, you can set $x_1=x_2 \equiv x$ in the correlation relation when deriving the Fokker-Planck equation. In the paper, the correlation relation is given in that more general form with $x_1$ and $x_2$ for other purposes than for derivation of the Fokker-Planck equation. From setting $x_1=x_2 \equiv x$ in the correlation relation follows $j_0(...)=1$ and thus we are left with a delta-correlation. The Fokker-Planck equation follows directly from the Kramers-Moyal expansion you have written there when you identify $\Gamma=f/\sqrt{H^3/8 \pi^2}$ and $g=\sqrt{H^3/8 \pi^2}$.

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