Reflection negative kernels and fractional Brownian motion
Summary: 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 \(\mathcal E\) and show in particular that fractional Brownian motion for Hurst index \(0 < H \leq 1 / 2\) 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 \(\mathrm{GL}_2(\mathbb{R})\). We relate this to a measure preserving action on a Gaussian \(L^2\)-Hilbert space \(L^2(\mathcal{E})\).
representations of \(\mathrm{SL}_2(\mathbb{R})\)
scientific article