Ergodic least-squares estimators of the generalize diffusion coefficient for fractional Brownian motion

Denis Boyer (1, 2) , David S. Dean (3) , Carlos Mejia-Monasterio (4, 5) , Gleb Oshanin (6)

1 Instituto de Fisica
2 Centro de Ciencias de la Complejidad
3 LOMA - Laboratoire Ondes et Matière d'Aquitaine
4 Laboratory of Physical Properties
5 Department of Mathematics and Statistics [Helsinki]
6 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

We analyze a class of estimators of the generalized diffusion coefficient for fractional Brownian motion Bt of known Hurst index H, based on weighted functionals of the single-time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true ensemble-average generalized diffusion coefficient to any necessary precision from single-trajectory data, but at the expense of a progressively higher experimental resolution. Convergence is fastest around H≃0.30, a value in the subdiffusive regime.

Mots clés: Probability theory; stochastic processes and statistics; Stochastic analysis methods; Single molecule kinetics; Single-molecule techniques

Domaines: Mécanique statistique [cond-mat.stat-mech]; Matière Molle [cond-mat.soft]

Denis Boyer, David S. Dean, Carlos Mejia-Monasterio, Gleb Oshanin. Ergodic least-squares estimators of the generalize diffusion coefficient for fractional Brownian motion. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2013, 87 (3), pp.030103. ⟨10.1103/PhysRevE.87.030103⟩. ⟨hal-00820816⟩