L\'{e}vy Laplacian for Square Roots of Measures

TL;DR

This paper redefines root measures in infinite-dimensional spaces, introduces calculus operations for them, and explores the Lévy Laplacian on Wiener space, linking its symbol to path quadratic variation.

Contribution

It develops a new framework for root measures, including differentiation, Fourier transform, and convolution, and applies it to analyze the Lévy Laplacian on Wiener space.

Findings

The symbol of the Lévy Laplacian equals the quadratic variation of paths.

Root measures are redefined within the context of infinite-dimensional measure theory.

New calculus operations for root measures are introduced and related.

Abstract

L.Accardi shows that the Banach space of singed measures is homeomorphic to the Hilbert space formed by so-called root measures. In this paper, we redefine root measures in view of the theory of measures on infinite dimensional spaces, and introduce a notion of differentiation, Fourier transform, and convolution product for root measures, and examine those relations. We also study about L\'{e}vy Laplacian on Wiener space as application. It is shown that the symbol of L\'{e}vy Laplacian is equal to the quadratic variation of paths.