Physical Brownian motion in a magnetic field as a rough path

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.