The Fokker-Planck equation for the Ornstein-Uhlenbeck reads
$$ \dot{p} = \gamma_{ij} \partial_{x_i}(x_jp)+D_{ij} \frac{\partial p}{\partial x_i\partial {x_j}}, $$ where $\gamma_{ij}, \, D_{ij}$ are constant matrices and the summation convention was used. I was trying to derive the general solution of this equation, which is given in my leture notes, without derivation, as $$ p(x,t) = \frac{1}{(2\pi)^{d/2}\sqrt{\det \Sigma}} \exp\left(-\frac{1}{2}(x-\langle x \rangle)^T \Sigma^{-1} (x-\langle x \rangle) \right), $$ where $\Sigma$ is the covariance matrix and $\langle x \rangle$ is the first moment of $p$. In my lecture notes it does not give the precise relations between ($\Sigma,\langle x \rangle)$ and $(\gamma_{ij},D_{ij})$ but I would like to know what they are and how to derive them.
I found this derivation (cf. section 3.2) which is basically copied from Risken's book (section 6.5) but there are a few steps I don't understand. Basically what is done there is:
- Substitute for p its Fourier transform to find the simpler (first order) equation:
$$\frac{\partial \tilde{p}}{\partial t} = - \gamma_{ij} k_i \frac{\partial \tilde{p}}{\partial k_j} - D_{ij} k_i k_j \tilde{p}. \tag{0}$$
Transform the initial condition $p(x,t'|x',t') = \delta(x-x')$ into Fourier space to find $$\tilde{p}(k,t'|x',t') = e^{-ikx'}$$
Since we are dealing with a diffusion equation (with drift), we expect the solution to be Gaussian. We make the ansatz: $$\tilde{p}(k,t|x',t') = \exp\left(-ik_l M_l (t-t')-\frac{1}{2}k_lk_m\sigma_{lm}(t-t')\right),$$ where $M_l$ and $\sigma_{lm}$ are functions of $t-t'$, which leads to one equation each for the real and imaginary part, (16,17)
- Solve each equation individually, noting that the initial condition translates to $M_i(t=0)=-x'$, $\sigma_{ij}(t=0)=0$.
- Transform back to x-space to find solution for p.
Now I am still trying to understand step 3. and 4. in particular. I understand (13), (14) and (16) from the reference but it looks to me as though there was a term missing in (15) which should come from product rule. I also find a different combination of indices in (17), namely $$\dot{\sigma_{ij}} = - \gamma_{il} (\sigma_{lj}+\sigma_{jl})+2D_{ij} \tag{1} $$ rather than Risken's equation, which is $$ \dot{\sigma_{ij}} = - \gamma_{il} \sigma_{lj} - \gamma _{jl} \sigma_{li})+2D_{ij} \tag{2}$$ Could someone please comment on how to get (2) and why (1) is wrong?
I have never solved an equation like (1) before and I would be glad about some advice about how to solve it. In particular, I substituted the solution (20) given in the reference but I don't see how it is supposed to solve (17) (nor (1)).
Alternatively, if someone could provide a solution of (0) using characteristics, I would be happy with that as well. I have tried the following. The characteristic equations read of (0) read 1) $\partial t/\partial s = 1 \Rightarrow t=s$, 2) $\partial k_i / \partial s = \gamma_{ji} k_j$, 3) $ \partial \ln(\tilde{p}) /\partial s = - D_{ij} k_{i}k_{j}$. Solving them gives $s=t$, $k_i(s) = \left[\exp(-\gamma^T s)\right]_{ij} k^0_j \equiv G_{ij} k^0_j$ and $\ln(\tilde{p}) = -\int D_{ij} G_{il} G_{jm}k^0_lk^0_m \mathrm{d} s$. Now I can exponentiate the last equation and obtain $\tilde{p}$. however, how do I Fourier transform back to position space?