A Classical Mechanical Model of Brownian Motion with One Particle Coupled to a Random Wave Field

Abstract

We consider the problem of deriving Brownian motions from classical mechanical systems. Specifically, we consider a system with one massive particle coupling to an ideal random wave field, evolved according to classical mechanical principles. We prove the almost sure existence and uniqueness of the solution of the considered dynamics, prove the convergence of the solution under a certain scaling limit and give the precise expression of the limiting process, a diffusion process.

Keywords:

• Brownian motion
• Classical mechanics
• Convergence
• Diffusions
• Wave field

AMS Classification (2010):

• 34F05
• 60B10
• 60J60
• 60J65

Acknowledgments

This work is financially supported by Grant-in-Aid for the Encouragement of Young Scientists (No. 21740063), Japan Society for the Promotion of Science.

The authors would like to thank Professor Sergio Albeverio for his careful reading of the manuscript.