Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

A support theorem for the X-ray transform on manifolds with plane covers

This paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

A SUPPORT THEOREM AND A LARGE DEVIATION PRINCIPLE FOR KUNITA FLOWS

In this paper, stochastic flows driven by Kunita-type stochastic differential equations are studied, focusing on support theorems (ST) and large deviation principles (LDP). We establish a new ST and LDP for Brownian flows with respect to a fine Hölder topology. Our approach is based on recent advances in rough paths theory, which is the natural framework for proving ST and LDP. Nevertheless, while rigorous, our presentation stays rather clear from the rough paths technicalities and is accessible for readers not familiar with them. We view the localized Brownian stochastic flow as a projection of the solution of a rough path differential equation implying the ST and LDP. In a second step the results are generalized for the global flow.