Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$
Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis??
Below is an image that represents examples of my problem. I would want the distribution of the total area colored red in both examples:
Progress thus far
I've come up with crude bounds based on hitting time distributions, based on a formula for the distribution of the first time a standard ($\sigma=1$) Brownian bridge on $[0,T]$ with endpoints $(0,0),(T,x)$ hits level $b$. I found the formula here (see top of page 19).
$$P(\tau_b\leq T|T,x)=e^{\frac{-2b(b-x)}{T}}$$
In the interests of simplicity, I started my analysis to just the standard brownian bridge, where $T=1,x=0$, this simplifies the formula to:
$$P(\tau_b\leq 1)=e^{-2b^2}$$
Ok, now let $A$ be the area between a standard brownian bridge and the x-axis. I know that $P(A\leq a) \geq P\left( \{\tau_{c}>1\} \cap \{\tau_{c-a}>1\}\right),\; 0\leq c \leq a$, which is simply the probability that the brownian bridge is contained inside a rectangle of area $a$ on $[0,T]$ such that it contains the x-axis. The $\leq$ is justified since $\sigma(\{\tau_{c}>1\} \cap \{\tau_{c-a}>1\})\subset \sigma(\{A\leq a\})$.
Now, there is nothing special about any particular choice of $c$ in the above, so I'd like to integrate the RHS of above over all valid choices of $c$, for this, I need the density of $P\left( \{\tau_{c}>1\} \cap \{\tau_{c-a}>1\}\right)$ as a function of $c$.
$$P\left( \{\tau_{c}>1\} \cap \{\tau_{c-a}>1\}\right) = P(\tau_{c-a}>1)P(\tau_{c}>1|\tau_{c-a}>1)=\left(1-e^{-2(c-a)^2}\right)P(\tau_{c}>1|\tau_{c-a}>1)=\left(1-e^{-2(c-a)^2}\right)\left(1-P(\tau_{c}\leq 1|\tau_{c-a}> 1)\right)$$
The conditional probability measure $P(\tau_{c}\leq 1|\tau_{c-a}> 1)$ stops me from getting anywhere with this approach.
Question
Can the conditional measure $P(\tau_{c}>1|\tau_{c-a}>1)$ be derived? Is there another route to getting the distribution function of the area bounded by the standard brownian bridge?