Consider the Ornstein-Uhlenbeck SDE
$$ dx = - \alpha x \, dx + \sqrt{2D} \, dW , x(0) = x_0 $$
It is known that
$$x(t) \sim N(x_0 e^{-\alpha t}, \frac{D}{\alpha}(1 - e^{-2\alpha t}))$$
and thus the transition probability density is given by
$$ p(x,t\mid x_0) = \sqrt{\frac{\alpha}{2 \pi D (1-e^{-2\alpha t})}} \exp \left(-\frac{\alpha(x-x_0e^{-2\alpha t})^2}{2D(1-e^{-2\alpha t})}\right) $$
Moreover, if $x_0 \sim p_0(x)$ for some initial density $p_0$ then
$$ p(x,t) = \int p(x,t\mid x_0) p(x_0) \, dx_0 $$
Now, my question is does anyone know where I might find solutions for $p(x,t)$ for various common densities for $p_0$?
For example, what is $p(x,t)$ when $p_0 = N(0,1)$?
One way to solve this is clearly just to integrate for $p(x,t)$ but this feels like the kind of thing that someone has done before and I just lack the vocabulary for to find it in a search.