I am trying to find the transition density of the following system of linear Langevin equations $$\begin{cases}dX_t=V_tdt\\dV_t=-X_tdt-\gamma V_tdt+\sqrt{2\gamma}dW_t\\(X(0),V(0))=(x_0,v_0)\end{cases}$$
My attempt: Write $$y=\begin{pmatrix}x\\v\end{pmatrix},b(x,v)=\begin{pmatrix}v\\-x-\gamma v\end{pmatrix},A(y)=\begin{pmatrix}0 & 0\\0 & 2\gamma\end{pmatrix}$$
The transition density satisfies the Fokker-Planck equation $$\partial_tp=-\partial_xvp+\partial_v(x+\gamma v)p+2\gamma\left(\partial_{xx}p+\partial_{vv}p\right)=0$$
I have found the stationary distrubution $p(x,v)=\exp\left(\frac{x^2-v^2}{2}\right)$, but I don't know how to proceed to get the transition density. My guess is that the transition density should involve the time variable $t$. Can anyone help me with this? Thank you.