I would like to ask to those, who have experience with PDE in stochastic process, for a bit of help. Here, I bring the Klein-Kramers equation which is a linear second-order PDE, given by
\begin{align} \frac{\partial \rho}{\partial t} = -\frac{P}{M}\left[\frac{\partial \rho}{\partial X}\right]+ M\omega^2 X \left[\frac{\partial \rho}{\partial P}\right] +\gamma\left[\frac{\partial \left(P\rho\right)}{\partial P}\right] +D\left[\frac{\partial^2 \rho}{\partial P^2}\right], \end{align}
where $\rho=\rho(X,P)$, $M$ is the particle mass, $\gamma$ is the dissipation factor, and $\omega$ is the harmonic oscillator frequency. Basically, the equation describes a Brownian particle in the phase space, i.e, in the space of position $X$ and momentum $P$. To get directly to the point, I want the solution of it, but so far I have not found it. In my attempt, I considered at first the Fourier transform, since it reduces the previous second-order differential equation to a first-order one. This is
\begin{align} \dot{R} = \frac{k_1}{m}\left[\frac{\partial R}{\partial k_2}\right] -M\omega^2 k_2 \left[\frac{\partial R}{\partial k_1}\right] - \gamma k_2\left[\frac{\partial R}{\partial k_2}\right] + Dk_2^2\,R, \end{align}
where $R=R(k_1,k_2,t) \equiv FT\left[\rho(X,P,t)\right]$. Then, I considered the method of characteristics using as initial condition that $\rho(X,P,0)=\delta(X)\delta(P)$, i.e., $R(k_1,k_2,0)=1$. However, I did not manage to find the solution. Actually, it would be already nice, if anyone knows how to get the solution of the differential equation above without the harmonic term, I mean:
\begin{align} \frac{\partial \rho}{\partial t} = -\frac{P}{M}\left[\frac{\partial \rho}{\partial X}\right] +\gamma\left[\frac{\partial \left(P\rho\right)}{\partial P}\right] +D\left[\frac{\partial^2 \rho}{\partial P^2}\right]. \end{align}