On joint returns to zero of Bessel processes

Abstract

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 $\hat{I}´\in (0,1)$, 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 $\hat{I}{¸}_n$ such that $\mathbb{P}(H_n>t)=t^{-\hat{I}{¸}_n+o(1)}$ as $t\to \infty$, and we provide some non-trivial bounds on $\hat{I}{¸}_n$. In particular, when $n=3$, we show that $2(1-\hat{I}´)\le \hat{I}{¸}_3\le 2(1-\hat{I}´)+f(\hat{I}´)$ for some (explicit) function $f(\hat{I}´)$ with ${\sup }_{[0,1]}f(\hat{I}´)\approx 0.079$.

26 pages, 3 figures, comments are welcome!