On joint returns to zero of Bessel processes

In this article, we consider joint returns to zero of $n$ Bessel processes ($n\geq 2$): 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 $(X_t^{(i)})_{1\leq i \leq n}$ of dimension $\delta \in (0,1)$, we are interested in the first joint return to zero of any two of them: $$H_n := \inf\big\{ t>0, \exists 1\leq i <j \leq n \text{ such that } X_t^{(i)} = X_t^{(j)} =0 \big\} \,.$$ We prove the existence of a persistence exponent $\theta_n$ such that $\mathbb{P}(H_n>t) = t^{-\theta_n+o(1)}$ as $t\to\infty$, and we provide some non-trivial bounds on $\theta_n$. In particular, when $n=3$, we show that $2(1-\delta)\leq \theta_3 \leq 2 (1-\delta) + f(\delta)$ for some (explicit) function $f(\delta)$ with $\sup_{[0,1]} f(\delta) \approx 0.079$.