Excursions from Hyperplanes for Isotropic Stable Processes

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Breaking the paths of isotropic Lévy processes in R in radial excursions from the origin is a technique that has been used to understand the behaviour of processes when they approach the origin or the unit sphere S ={formulae = 1 }. We aim to build on these ideas and techniques and consider related behaviour in relation to the hyperplane H = {formulae =0 }, with v / S, and use the associated path decomposition to address questions such as how to condition the process to stay on one side of the hyperplane, examining the point of closest reach to H, conditioning to never touch the hyperplane or the relationship between this and the spherical case via conformal transformations of Riemannian manifolds.