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Symmetry 2018, 10(6), 191; https://doi.org/10.3390/sym10060191

Reflection Negative Kernels and Fractional Brownian Motion

1
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
2
Department Mathematik, FAU Erlangen–Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
3
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
*
Author to whom correspondence should be addressed.
Received: 7 May 2018 / Revised: 20 May 2018 / Accepted: 20 May 2018 / Published: 1 June 2018
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
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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 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 ) . View Full-Text
Keywords: fractional brownian motion; reflection positivity; reflection negative kernels; representations of S L 2 ( R ) fractional brownian motion; reflection positivity; reflection negative kernels; representations of S L 2 ( R )
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Jorgensen, P.E.T.; Neeb, K.-H.; Ólafsson, G. Reflection Negative Kernels and Fractional Brownian Motion. Symmetry 2018, 10, 191.

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