Year: 2015
Type: article
Abstract: The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One can scale these models by changing the mass of the particle, and in the small mass limit one has almost sure uniform convergence in distribution to the standard idealized model of mathematical Brownian motion. This provides one well-known way of realising the Wiener process. However, this result is less robust than it would appear, and important generic functionals of the trajectories of the physical Brownian motion do not necessarily converge to the same functionals of Brownian motion when one takes the small mass limit. In the presence of a magnetic field the area process associated to the physical process converges - but not to Lévy’s stochastic area. As this area is felt generically in settings where the particle interacts through force fields in a non-linear way, the remark is physically significant and indicates that classical Brownian motion, with its usual stochastic calculus, is not an appropriate model for the limiting behaviour. We compute explicitly the area correction term and establish convergence, in the small mass limit, of the physical Brownian motion in the rough path sense. The small mass limit for the motion of a charged particle in the presence of a magnetic field is, in distribution, an easily calculable, but “non-canonical” rough path lift of Brownian motion. Viewing the trajectory of a charged Brownian particle with small mass as a rough path is informative and allows one to retain information that would be lost if one only considered it as a classical trajectory. We comment on the importance of this point of view.
Authors: Peter Friz, Paul Gassiat, Terry Lyons
Institutions: Weierstrass Institute for Applied Analysis and Stochastics, Technische Universität Berlin, Centre National de la Recherche Scientifique, Centre de Recherche en Mathématiques de la Décision, Université Paris Cité, Université Paris Diderot, Université Paris Dauphine-PSL, University of Oxford
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: bronze
Awards: G2121218313, G2902570791, Increasing the efficiency of numerical methods for estimating the state of a partially observed system. High order methods for solving parabolic PDEs, G3207356259, G3380566117, Rough path theory, differential equations and stochastic analysis, Creating rigorous mathematical and computational tools that can summarise high dimensional data streams in terms of their effects, G393177772, G5414291753, G5702163051, G5878972985, G7326578543
OpenAlex ID: W1510680323