On joint returns to zero of Bessel processes

In this article, we consider joint returns to zero of $n$ Bessel processes ($n\ge 2$): our main goal is to estimate the probability that they avoid having joint returns to zero for a long time. More precisely, considering $n$ independent Bessel processes $(X_t^{(i)}{)}_{1\le i\le n}$ of dimension $\delta \in (0,1)$, we are interested in the first joint return to zero of any two of them: $$H_n:=\inf \{t>0,\mathrm{\exists }1\le i<j\le n\text{ such that }{X}_t^{(i)}=X_t^{(j)}=0\}\mathrm{.}$$ We prove the existence of a persistence exponent ${\theta }_n$ such that $\mathbb{P}(H_n>t)=t^{-{\theta }_n+o(1)}$ as $t\to \mathrm{\infty }$, and we provide some non-trivial bounds on ${\theta }_n$. In particular, when $n=3$, we show that $2(1-\delta )\le {\theta }_3\le 2(1-\delta )+f(\delta )$ for some (explicit) function $f(\delta )$ with ${\sup }_{[0,1]}f(\delta )\approx 0.079$.