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We review and study a one-parameter family of functional transformations,denoted by $(S^{(\beta)})_{\beta\in \R}$, which, in the case $\beta<0$,provides a path realization of bridges associated to the family of diffusionprocesses enjoying the time inversion property. This family includes theBrownian motion, Bessel processes with a positive dimension and theirconservative $h$-transforms. By means of these transformations, we derive anexplicit and simple expression which relates the law of the boundary crossingtimes for these diffusions over a given function $f$ to those over the image of$f$ by the mapping $S^{(\beta)}$, for some fixed $\beta\in \mathbb{R}$. We givesome new examples of boundary crossing problems for the Brownian motion and thefamily of Bessel processes. We also provide, in the Brownian case, aninterpretation of the results obtained by the standard method of images andestablish connections between the exact asymptotics for large time of thedensities corresponding to various curves of each family.

Boundary crossing identities for diffusions having the time inversion property