This question comes from book [Ju-Yi Yen and Marc Yor](https://link.springer.com/book/10.1007/978-3-319-01270-4) [P59](https://i.stack.imgur.com/DJGja.png) and [P60](https://i.stack.imgur.com/3C4qD.png),


On page 59, "Define $\mathcal{Z}\_\omega=\{t:B\_t(\omega)=0\},$ and $\tau\_l$ is the inverse local time. The complement of $\mathcal{Z}\_\omega$is shown to be $\mathcal{Z}\_\omega^c=\bigcup\_{s>0}\left]\tau\_{s-},\tau\_s\right[$, where $]\tau\_{s-},\tau\_s[$ are the maximal intervals of constancy of the local time $L$. This defines the excursion process as $e\_l(\omega)=B(\tau\_{l-}+t) 1\_{(t\leq \tau\_s-\tau\_{s-})}$"


On page 60, "$(e\_s, s\geq 0)$ is a Poisson point process."


My question is:


1. how to understand the definition of excursion process $e\_l(\omega)=B(\tau\_{l-}+t) 1\_{(t\leq \tau\_s-\tau\_{s-})}$? what is the picture?
2. how to prove this excursion process is a Poisson Point Process?



How to prove excursion process is a Poisson point process?
The standard reference for Brownian excursions is Chapter 12 in the classic book [1]. Other developments of excursion theory can be found in [2]-[4] and many other sources.


To develop the intuition, a good approach is to start with random walks.
(See, e.g., the exposition of local time in [6]).
The counting measure on the zero set $\{Z\_0=0, Z\_1, Z\_2,\ldots \}$ of the symmetric simple random walk (SRW) on the integers,
suitably normalized, converges to the Brownian local time at zero. Hence it is natural to understand the excursions of SRW of duration at least $2\ell$ that start in one of $Z\_{an},\dots, Z\_{bn}$. Using the reflection principle and Stirling, see e.g. [5] Thm 9.3, we know that the chance a SRW excursion will be "long" (=have duration greater than $2\ell$) is asymptotic to $p\_\ell=(\pi \ell)^{-1/2}$ as $\ell$ grows. Moreover, these events are independent for each of the excursions started at $Z\_{an},\dots, Z\_{bn}$, so the total number of "long" excursions among these has a Binomial Bin$(bn-an,p\_\ell)$ distribution. To get a nontrivial limit as $n,\ell \to \infty$, take $\ell=\ell(n)=tn^2$ so that $p\_\ell=(\pi t)^{-1/2}/n$. We deduce that the number of "long" excursions (of duration $\ge 2tn^2$) started in $Z\_{an},\dots, Z\_{bn}$ has a limit distribution $$\lim\_n { \rm Bin}(bn-an,(\pi t)^{-1/2}/n)={\rm Poisson}((b-a) \cdot (\pi t)^{-1/2})\,.$$ These counts of long excursions are independent for disjoint intervals
$[a\_1,b\_1], [a\_2,b\_2],\ldots$, so the counting measure of long excursions will indeed converge to a Poisson process as $n \to \infty$.


[1] Revuz, Daniel, and Marc Yor. Continuous martingales and Brownian motion. Vol. 293. Springer Science & Business Media, 2013.


[2] Pitman, J., and M. Yor. "Itô's excursion theory and its applications." Japanese Journal of mathematics 2, no. 1 (2007): 83.


[3] Watanabe, S. (2010). Itô’s theory of excursion point processes and its developments. Stochastic processes and their applications, 120(5), 653-677.
<https://core.ac.uk/download/pdf/82811223.pdf>


[4] K. L. Chung, Excursions in Brownian motion, Ark. Mat., 14 (1976), 155–177.


[5] Révész, Pál. Random walk in random and non-random environments. World Scientific, 2013.


[6] Mörters, Peter, and Yuval Peres. Brownian motion. Vol. 30. Cambridge University Press, 2010.


