This question was asked by another user, but was deleted. As it may be useful for others, I re-post it here.

What are the transition distribution (or density) functions of two processes defined by \begin{align*} dX_t = \mu dt + \sigma dW_t \end{align*} and \begin{align*} dX_t = \theta(\mu-X_t) dt + \sigma dW_t, \end{align*} where $\theta>0$, $\mu$ is a real number, $\sigma >0$, and $\{W_t,\, t \ge 0\}$ is a standard Brownian motion.