The joint distribution of zeros of brownian motion Let $\gamma_t$ be the last zero of brownian motion before $t$ and $\beta_t$ be the first zero after $t$. I need to calculate the joint distribution of $\gamma_t$ and $\beta_t$, i.e. $P(\gamma_t<x, \beta_t<y)$.
I know about arcsine low for these zeroes:
$P(\gamma_t<x)=\frac{2}{\pi}\arcsin\sqrt\frac{x}{t}$
$P(\beta_t<y)=\frac{2}{\pi}\arcsin\sqrt\frac{t}{y}$
But how can I get the joint distribution?
Thanks!
 A: Let me assume for a moment that one knows the distribution of the first hitting time $\tau$ of level $1$ by a Brownian motion starting from $0$. By scaling and translation invariance, the first hitting time of $0$ for a Brownian motion starting from $x$ is distributed like $x^2\tau$.
For every $0\lt u\lt t\lt v$, $[\gamma_t\lt u,v\lt\beta_t]=[\gamma_v\lt u]$. The scale invariance of Brownian motion implies that $[\gamma_v\lt u]$ has the same probability as $[\gamma_1\lt u/v]$. The last event depends on $\gamma_1$ only, and one knows that the distribution of $\gamma_1$ is the Arcsine distribution. Let us briefly show this.
Conditionally on $B_{s}=x$, the event $[\gamma_1\lt s]$ corresponds to the fact that a Brownian motion starting from $x$ hits $0$ after time $1-s$, hence
$$
\mathrm P(\gamma_1\lt s\mid B_{s}=x)=\mathrm P(x^2\tau\gt 1-s).
$$
Introducing $h(s)=\mathrm P(\tau\gt s)$ for every $s\gt0$ and writing $(g_s)_{s\gt0}$ for the transition semi-group of Brownian motion $g_s(x)=\mathrm e^{-x^2/(2s)}/\sqrt{2\pi s}$, this means that
$$
\mathrm P(\gamma_1\lt s)=\int_{-\infty}^{+\infty}h\left(\frac{1-s}{x^2}\right)g_s(x)\mathrm dx,
$$
from which the distribution of $\gamma_1$, thus also the joint distribution of $(\gamma_t,\beta_t)$, follow by differentiation.
To complete this, recall that Désiré André's reflexion principle shows that the probability of $[\tau\lt s]$ is twice the probability of $[B_s\gt1]$, hence
$$
h(s)=1-2\mathrm P(B_s\gt 1)=2\int_0^1g_s(x)\mathrm dx.
$$
After some simplifications and change of variables in the double integral, this leads to
$$
\mathrm P(\gamma_1\lt s)=\frac2\pi\arctan\left(\sqrt{\frac{s}{1-s}}\right)=\frac2\pi\arcsin\left(\sqrt{s}\right),
$$
hence
$$
\mathrm P(\gamma_t\lt u,v\lt\beta_t)=\frac2\pi\arcsin\left(\sqrt{\frac{u}{v}}\right),
$$
a formula which is all over the place, and probably in Rick Durrett's Probability Theory with Applications.
