Sharp barrier estimates for Bessel bridges
Leandro Chiarini, Ellen Powell
TL;DR
This work delivers sharp barrier estimates for a $d$-dimensional Bessel bridge relative to an affine barrier, providing precise asymptotics for the constrained path probability as the time horizon grows. The authors reduce the affine-barrier problem to a flat-barrier problem through time-inversion and scaling, then obtain a leading-term expression in terms of $\mathcal{P}_{a,b,x}$ and $\widehat{\mathcal{P}}_b(x)$ plus a vanishing error, and they establish non-sharp universal bounds as well. They further explore the behaviour as $a$ and $j$ grow and prove a perturbation result for small concave barrier corrections $h$, yielding a limsup bound with a vanishing relative error. The results illuminate the maxima and extremal behavior of log-correlated systems and have potential applications to models like branching Brownian motion and the planar Gaussian free field, where constrained path probabilities underpin fine-scale extreme value analysis.
Abstract
In this article, we derive precise estimates for the probability that a Bessel bridge of dimension $d \ge 0$ and end points $x$ and $a+bT-j$ stays below the linear barrier $a + bt$ for all $t \in [0,T]$. We identify the leading order term as well as the asymptotic error for this probability as $T\to \infty$, depending on $a,b,j,x$. We also derive the behaviour of such leading term as we allow $a,j\to \infty$, and obtain precise bounds for all error terms. Finally, we establish a complementary result where the linear barrier is perturbed by a small concave function.