Let $W_t$ be the Wiener process (Brownian motion). I am interested in the following distributions/probabilities
- What is the expected number of intervals on $[0,1]$ on which $W_t<a$, where $a<0$ is a given number? More precisely: since $W_t$ is continuous in $t$, the set $U=\{t\in[0,1]:W_t<a\}$ is open. Therefore, $U$ can be expressed as at most countably many open intervals. Write $U=\bigcup_{k=1}^n B_k$, where $B_k$ are disjoint open intervals. We may have $n=\infty$. My question is: what is the expected value of $n$? What distribution does $n$ follow?
- Let $m$ be number of intervals in $\{B_k\}$ with length at least $b$. $m$ is always finite. What is the expected value of $m$?
I do not find these in my random process book, but I think they are quite natural questions to ask. I would really appreciate it, if anyone could provide me with some reference about those problems, or write an answer to them.