I am working on Langevin equations:
$\frac{dx}{dt}=u$
$m\frac{du}{dt}= -\gamma u + \theta(t)$
where $\theta(t)$ is delta-correlated in time Gauss-distributed noise with zero-mean
$\langle \theta (t) \rangle = 0 $
$\langle \theta (t) \theta (t') \rangle = 2 D \delta(t-t') $
and $\gamma=const$ and $D=const$ are constant coefficients.
I can find the probability of $x$ (position) and probability of $u$ (velocity) given initial position $x_0$ and velocity $u_0$ (see for example [1]):
$F(x,t|u_0,x_0)=\frac{1}{Z_F}\exp\Big[\frac{m \gamma}{2D} \frac{(x-x_0-u_0 (1- e^{-\gamma t})/\gamma)^2}{2\gamma t - 3 + 4 e^{-\gamma t} - e^{-2\gamma t}} \Big]$
$G(u,t|u_0,x_0)=\frac{1}{Z_G}\exp\Big[\frac{m \gamma}{2D} \frac{(u-u_0 e^{-\gamma t})^2}{1 - e^{-2\gamma t}} \Big]$
where $Z_F$ and $Z_G$ are some normalization coeffitients.
But I am missing a joint-probability of $x$ and $u$, meaning a probability for particle to have simultaneously position $x$ and velocity $u$ given $x_0$ and $u_0$ or $P(x,u,t|u_0,x_0)$. Any help to derive $P(x,u,t|u_0,x_0)$ exactly or approximately would be highly appreciated.
[1] - http://dx.doi.org/10.1103/PhysRev.36.823, G. E. Uhlenbeck and L. S. Ornstein, 1930