The classical diffusion equation is initially a Fokker-Planck equation. Such equation could be reduced back to its corresponding Langevin equation.
$\partial_tP(x,t) = \partial_{xx}[ D(x)P(x,t) ] $
$\partial_t x(t) = \Gamma(x,t) \\ $
where $\langle \Gamma(x,t) \rangle = 0$, $\langle \Gamma(x,t) \Gamma(x,s) \rangle = S(x)\delta(t-s)$ and $S(x)$ characterises the local diffusivity $D(x)$.
Instead, I am interested in simulating the fractional Brownian motion (fBm) with $\langle \eta(x,t) \rangle = f(x)$, i.e. position-dependent mean of noise.
$\partial_t x(t) = \eta(x,t) \\ $
However, the expression of fractional Gaussian noise $\eta$ is more difficult to find. Usually, people simulate the fractional Brownian motion directly.
In this case, is it possible to proceed on with the Langevin equation directly? Since the noise is now time-correlated, is it still possible to perform Kramers-Mayal expansion to obtain its Fokker-Planck equation?