Generalized Radon Transform and Lévy’s Brownian Motion, I

Abstract

In connection with a Gaussian system X = {X(x); x ∈ M} called Lévy’s Brownian motion (Definition 1), we shall introduce two integral transformations of special type—one is a generalized Radon transform R on a measure space (M, m), and the other is a dual Radon transform R* on another measure space (H, v) such that H ⊂ 2M, the set of all subsets of M (Definition 2). To each Lévy’s Brownian motion X, there is attached a distance d(x, y):= E[(X(x) — X(y)2] on M having a notable property named L1-embeddability. The above measure v on H is then chosen to satisfywhere Δ stands for the symmetric difference.