## Abstract

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass μ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity ϵ> 0. We show that for all small but fixed frictions the small mass limit of q_{μ}_{,}_{ϵ} gives the solution q_{ϵ} to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion q_{ϵ} and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

Original language | English (US) |
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Pages (from-to) | 132-148 |

Number of pages | 17 |

Journal | Journal of Statistical Physics |

Volume | 181 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2020 |

## Keywords

- Averaging principle
- Hamiltonian systems
- Smoluchowski–Kramers approximation
- Stochastic differential equations
- Stochastic equations on graphs

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics