Consider the following OU process parameterized by two positive real numbers $\alpha$ and $\sigma$:
$$dX_t = -\alpha X_t dt + \sigma \sqrt{2\alpha} dW_t$$ $$X_0 = x_o > 0$$
We are interested in the first passage time $\tau = \inf \left\lbrace t \geq 0: X_t = 0\right\rbrace$. From what I read, the density function for the first passage time is given by:
$$f(t) = \frac{x_0}{2\sigma\sqrt{\pi \alpha}}\exp\left( -\frac{x_0^2e^{-\alpha t}}{4\sigma^2\sinh(\alpha t)}+\frac{\alpha t}{2}\right)\left(\frac{\alpha}{\sinh(\alpha t)}\right)^{3/2}$$
Where does this expression come from? The only place I can find any semblance of a proof is here, but I don't really follow it completely. So I'm curious if there's a bit of a more elementary derivation of this equation, something geared more towards someone with general knowledge in this field but not someone who is an expert.