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Reflection negative kernels and fractional Brownian motion

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arxiv 1805.02593 v1 pith:ET6FM327 submitted 2018-05-07 math-ph math.MPmath.PR

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keywords reflectionbrownianfractionalmotionhilbertpositivityspacerelate
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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).

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