$X_t$ is a weak solution to the SDE with $dX_t = ( −\alpha X_t + \gamma)dt + \beta dB_t$ , $\forall t \geq 0$
$X_0 = x_0$. $\;\;\;\alpha$, $\beta$, and $\gamma$ constants, and $Bt$ is a Brownian motion.
I need to find the PDE for the transition density of $X$ at time $t$, $pt(x_0,.)$, and solve it
the expectation is $E[X_t]=\gamma / \alpha + \exp(-\alpha t)(x_0-\frac{\gamma}{\alpha})$ and the variance $V[X_t]=\frac{b^2}{2\alpha}(1-\exp(-2αt))$.
and I know that $X_t$ follows a normal distribution and that it has a stationary distribution. so $p(t,x,y)=N~(E[X_t],V[X_t])$
but how to find the PDE of the transition density/ probability distribution?
