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Physical Brownian motion in a magnetic field as a rough path

Authors: Peter Friz, Paul Gassiat and Terry Lyons
Journal: Trans. Amer. Math. Soc. 367 (2015), 7939-7955
MSC (2010): Primary 60H99
Published electronically: March 24, 2015
MathSciNet review: 3391905
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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.

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Additional Information

Peter Friz
Affiliation: Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany — and — Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, 10117 Berlin, Germany

Paul Gassiat
Affiliation: UFR Mathématiques, University of Paris Diderot 7, 85205, Paris Cedex 13, France
Address at time of publication: CEREMADE, Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France

Terry Lyons
Affiliation: Oxford Man Institute, University of Oxford, Eagle House, Walton Well Road, Oxford OX2 6ED, United Kingdom

Keywords: Physical Brownian motion, homogenization, L\'evy's area
Received by editor(s): February 4, 2013
Received by editor(s) in revised form: August 20, 2013
Published electronically: March 24, 2015
Additional Notes: The research of the first and second authors was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr. 258237. The research of the third author was supported by the Oxford-Man Institute and the European Research Council under the European Union’s Seventh Framework Programme (FP7-IDEAS-ERC) / ERC grant agreement nr. 291244.
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