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 ε 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 GL2(ℝ). We relate this to a measure preserving action on a Gaussian L2-Hilbert space L2(ε).
Publication Source (Journal or Book title)
Jorgensen, P., Neeb, K., & Ólafsson, G. (2018). Reflection negative kernels and fractional Brownian motion. Symmetry, 10 (6) https://doi.org/10.3390/sym10060191