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Brownian windings, Stochastic Green's formula and inhomogeneous magnetic impurities

Isao Sauzedde

TL;DR

This work develops a robust stochastic Green formula for planar Brownian motion, assigning a principal value to the otherwise singular winding-weighted integral and proving a Green-type relation between a Stratonovich line integral and a winding-weighted two-form. The construction hinges on symmetric cut-offs of the winding function and dyadic decompositions to control singularities, yielding almost sure convergence and a continuous linear functional on a Hölder-Sobolev space of forms. The paper then applies this framework to a physically motivated model of Aharonov–Bohm phases with inhomogeneous magnetic impurities, deriving a 1-stable central limit theorem: as the impurity density grows, the averaged winding converges in distribution to a deterministic drift plus a Poissonian 1-stable noise term, with the drift determined by the inhomogeneous weight and the geometry of the Brownian path. These results connect stochastic Green formulas, winding theory, and random media, providing explicit limiting laws and a rough-path interpretation of environmental randomness in planar diffusion.

Abstract

We give a general Green formula for the planar Brownian motion, which we apply to study the Aharonov--Bohm effect induced by Poisson distributed magnetic impurities on a Brownian electron in the presence of an inhomogeneous magnetic field.

Brownian windings, Stochastic Green's formula and inhomogeneous magnetic impurities

TL;DR

This work develops a robust stochastic Green formula for planar Brownian motion, assigning a principal value to the otherwise singular winding-weighted integral and proving a Green-type relation between a Stratonovich line integral and a winding-weighted two-form. The construction hinges on symmetric cut-offs of the winding function and dyadic decompositions to control singularities, yielding almost sure convergence and a continuous linear functional on a Hölder-Sobolev space of forms. The paper then applies this framework to a physically motivated model of Aharonov–Bohm phases with inhomogeneous magnetic impurities, deriving a 1-stable central limit theorem: as the impurity density grows, the averaged winding converges in distribution to a deterministic drift plus a Poissonian 1-stable noise term, with the drift determined by the inhomogeneous weight and the geometry of the Brownian path. These results connect stochastic Green formulas, winding theory, and random media, providing explicit limiting laws and a rough-path interpretation of environmental randomness in planar diffusion.

Abstract

We give a general Green formula for the planar Brownian motion, which we apply to study the Aharonov--Bohm effect induced by Poisson distributed magnetic impurities on a Brownian electron in the presence of an inhomogeneous magnetic field.
Paper Structure (16 sections, 20 theorems, 144 equations)

This paper contains 16 sections, 20 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Stochastic Green's formula
  3. Magnetic impurities
  4. Notations
  5. Differential forms and integrals
  6. Winding
  7. Cauchy variables
  8. Former results
  9. Stokes formula
  10. Existence of a limit
  11. Strategy for the Green formula
  12. Proof of Lemma \ref{['le:line']}
  13. Proof of Lemma \ref{['le:residu']}
  14. Proof of Lemma \ref{['le:strato']}
  15. Magnetic impurities
  16. ...and 1 more sections

Key Result

Theorem 1

Let $X:[0,T]\to \mathbb{R}^2$ be a Brownian motion, and let ${\mathbf n}_X$ be the winding function associated with the loop obtained by concatenation of $X$ with the straight line segment $[X_T,X_0]$ between its endpoints. Then, almost surely, for all $f\in \mathcal{C}_2^{\beta}(\mathbb{R}^2)$ for converges as $N\to \infty$. Define ${ \vcenter{\hbox{$-$}}} \!\int_{\mathbb{R}^2} {\mathbf n}_X

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Remark 6
  • Remark 7
  • Lemma 3.1: LAWA
  • Lemma 3.2: LAWA
  • Lemma 3.3: BWE
  • ...and 24 more