I'm looking for functions $g: \mathbb{R}_+ \to \mathbb{R}$ such that the hitting time
$$\tau := \inf \{t \geq 0 : B_t \nleq g(t) \} $$
has an explicit density with respect to the Lebesgue measure, where $B_t$ is a standard Brownian motion. What are possible examples?
The case where, for some $R> 0$, $g(t) \equiv R$ is constant leads to the hitting time having Lévy distribution. Indeed for $g(t) = R - \alpha t$, we also obtain an explicit density of $$\rho_{\tau}(t) = \frac{R}{\sqrt{4\pi t^3}} e^{-\frac{(R - \alpha t)^2}{4t}}$$
Are there other functions out there where a similar representation exists? Is there an explicit density for, say, $\inf \{t \geq 0 : B_t \nleq M_t \}$ for some process $M_t$ with $M_0 > 0$.
