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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<Hle 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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Dynamics of quantum systems which are perturbed by linear coupling to the reservoir stochastically can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation). To work it out one needs definition of quantum Brownian motion. Since till very recent times only its boson version has been known, in present paper we demonstrate definition which makes possible consideration also for fermion Brownian motion.
We give a stochastic proof of the finite approximability of a class of Schru007fodinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.
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We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.
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