If $W_t$ is a standard Wiener process, Levy's Arcsine Law gives us the CDF of the random variable
$$ \int_0^1 \delta[W_t \ge 0] dt $$
(where $\delta[\text{event}]$ is $1$ if the event happens and $0$ otherwise) as $\frac{2}{\pi}\arcsin(\sqrt{x}), x\in[0,1]$. I am aware there is a generalization to $W(t)\ge\alpha$ instead of just $W(t)\ge 0$, and also a generalization to Brownian Motion with drift (i.e. $W_t+\mu t$ where $\mu$ is some constant).
Is there a similar formula for Brownian Bridges? I.E. if $B_t, t\in[0,1]$ is a Brownian Bridge from $B_0=0$ to $B_1=b$ (the simplest case would be $b=0$, but hopefully we can generalize to non-horizontal bridges), can we say anything about the distribution of the following random variable?
$$ \int_0^1 \delta[B_t \ge \alpha] dt $$
I know if $b=0,\alpha=0$ then the distribution is uniform. There might be a way to transform the drifted Brownian motion result into a Brownian Bridge result, but I couldn't think of such an idea. (In particular, the Bridge distribution for $b=0, \alpha=0$ does not coincide with the Motion distribution for $\mu=0, \alpha=0$).
Potential approach: We study the occupation time $T_{r}:=\int_{0}^{1}1_{B_{t}> r}dt$ following the notes Arcsin Laws and Feynman-Kac for Brownian bridge. As mentioned in Transition density of Brownian bridge using generators, the generator is
$$A(f)=\frac{-x}{1-t}f'+\frac{1}{2}f''$$
and the transition density satisfies
$$-\frac{x}{1-t}\frac{\partial }{\partial x}p(t,x)+\frac{1}{2}\frac{\partial^{2} }{\partial^{2} x}p(t,x)=\frac{\partial }{\partial t}p(t,x).$$
So following Arcsin Laws and Feynman-Kac, we have the analogous equation
$$u_{t}=\left\{\begin{matrix} \frac{1}{2}u_{xx}-\frac{x}{1-t}u_{x}-\lambda u&x> r\\\frac{1}{2}u_{xx} -\frac{x}{1-t}u_{x}&x\leq r \end{matrix}\right.$$
Using again the Laplace transform $\hat{u}(\alpha,x)$, we get a PDE system: for $x> r$
$$-1+\alpha\hat{u}(\alpha,x)+\frac{d\hat{u}(\alpha,x)}{d\alpha}= \frac{1}{2}(\hat{u}_{xx}(\alpha,x)+\frac{d\hat{u}_{xx}(\alpha,x)}{d\alpha})-x\hat{u_{x}}(\alpha,x)-\lambda (\hat{u}(\alpha,x)+\frac{d\hat{u}(\alpha,x)}{d\alpha}) ,$$
and $x\leq r$ $$-1+\alpha\hat{u}(\alpha,x)+\frac{d\hat{u}(\alpha,x)}{d\alpha}= \frac{1}{2}(\hat{u}_{xx}(\alpha,x)+\frac{d\hat{u}_{xx}(\alpha,x)}{d\alpha})-x\hat{u_{x}}(\alpha,x).$$
Since this is again a PDE of similar form, it is unclear if this can be solved explicitly. If anybody has some suggestions, I will be happy to try them out.