Journal Article

Random Fourier Series and Continuous Additive Functionals of Levy Processes on the Torus

Rosen, Jay, Marcus, Michael B.

1996

Overview

Let X be an exponentially killed Levy process on Tn, the n-dimensional torus, that satisfies a sector condition. (This includes symmetric Levy processes.) Let Fedenote the extended Dirichlet space of X. Let h ∈ Feand {hy, y ∈ Tn} denote the set of translates of h. That is, hy(·) = h(· - y). We consider the family of zero-energy continuous additive functions {N[ hy] t, (y, t) ∈ Tn× R+} as defined by Fukushima. For a very large class of random functions h we show that$J_\\rho(T^n) = \\int(\\log N_\\rho(T^n, \\varepsilon))^{1/2} d\\varepsilon < \\infty$is a necessary and sufficient condition for the family {N[ hy] t, (y, t) ∈ Tn× R+} to have a continuous version almost surely. Here Nρ(Tn, ε) is the minimum number of balls of radius ε in the metric ρ that covers Tn, where the metric ρ is the energy metric. We argue that this condition is the natural extension of the necessary and sufficient condition for continuity of local times of Levy processes of Barlow and Hawkes. Results on the bounded variation and p-variation (in t) of N[ hy] t, for y fixed, are also obtained for a large class of random functions h.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 42A61 / 60G15 / 60J45 / 60J55 / Continuous additive functionals / Dirichlet spaces / Exact sciences and technology / Fourier analysis / Fourier coefficients / Fourier series / Global analysis, analysis on manifolds / Markov processes / Mathematical analysis / Mathematical theorems / Mathematics / Perceptron convergence procedure / Probabilities / Probability and statistics / Probability theory and stochastic processes / random Fourier series / Random variables / Sciences and techniques of general use / Sequences, series, summability / Series convergence / Stochastic analysis / Stochastic processes / Sufficient conditions / Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds