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 $M\_0 > 0$.



Explicit densities for Brownian motion hitting times
See e.g. [ANALYTIC CROSSING PROBABILITIES FOR CERTAIN
BARRIERS BY BROWNIAN MOTION](https://arxiv.org/pdf/0704.2826.pdf), especially Example 2 (page 9) there for a square-root barrier, Example 3 (page 9) there for a square-root-times-log-factor barrier, and further references therein. 


