Reflection negative kernels and fractional Brownian motion

TL;DR

This paper explores the mathematical connection between fractional Brownian motion, reflection positivity, and representation theory, revealing new insights into quantum physics and infinite-dimensional Hilbert spaces.

Contribution

It introduces reflection positivity for affine isometric actions and demonstrates its application to fractional Brownian motion with Hurst index 0<H≤1/2.

Findings

Fractional Brownian motion with 0<H<1/2 is reflection positive.

Reflection positivity leads to an infinite-dimensional Hilbert space.

Relation between fractional Brownian motion invariance and complementary series representations.

Abstract

In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0<H\le 1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0<H <1/2. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL(2,R). We relate this to a measure preserving action on a Gaussian L^2-Hilbert space L^2(E).