Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $ dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0} $ (1) Let $p_{t}(x_{0},\cdot)$ be the transition density for $X$ at time $t$. Find the partial differential equation (PDE) for $p_{t}\left(x_{0},\cdot\right)$ and solve.
(2) Does $X_{t}$ have a stationary distribution? and if so find it.
(3) Using stochastic methods find explicit solution to each of the two: $i=1,\,2$ initial value problems: $ \partial_{t}u(t,x)=\frac{1}{2}\beta^{2}\partial_{xx}^{2}u(t,x)+\left(-\alpha x+\gamma\right)\partial_{x}u(t,x), $ and $u(0,x)=f_{i}(x)$ where $f_{1}(x)=\delta_{x^{*}}(x)$ is the Dirac function ($\delta_{x^{*}}(x)=1$ if $x=x^{*}$, $\delta_{x^{*}}(x)=0$ if $x\neq x^{*}$), and $f_{2}(x)=x$.
I came accross the above problem while preparing for my SDE exam. It was on a past paper. I would be grateful to someone who can clearly explain to me the solution process. :)