Title: Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes

 Abstract: We are interested in the law of the first passage time of driftless Ornstein-Uhlenbeck processes to time varying thresholds. We show that this problem is connected to the law of the first passage time of the process to some two-parameter family of functional transformations which, for specific values of the parameters, appears in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first proof is based on a similar result to the case of the Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method applied to the Fokker-Planck equation of the Ornstein-Uhlenbeck process. We investigate the properties of this transformation and study the algebraic and analytical properties of an involution operator which is used in constructing it. We also show that this transformation maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we discuss the interpretation of such transformations through the method of images and give new examples of curves with explicit first passage time densities.