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True self-repelling motion above a general barrier

Laure Marêché

Abstract

The true self-repelling motion is a continuous-time random process which was introduced by Tóth and Werner in 1998 to be a limit for the "true" self-avoiding random walk defined by Tóth in 1995. The construction of the true self-repelling motion involves an uncountable system of coalescing Brownian motions starting from all points of the upper half-plane, related to the Brownian web, but reflected and absorbed on a "barrier" which is the abscissa axis. In this work, we consider much more general barriers, construct an uncountable system of coalescing Brownian motions reflected and absorbed on these barriers, and the true self-repelling motion associated to it. The extension of the proofs of Tóth and Werner to this more general case is surprisingly difficult, especially when the barrier is irregular.

True self-repelling motion above a general barrier

Abstract

The true self-repelling motion is a continuous-time random process which was introduced by Tóth and Werner in 1998 to be a limit for the "true" self-avoiding random walk defined by Tóth in 1995. The construction of the true self-repelling motion involves an uncountable system of coalescing Brownian motions starting from all points of the upper half-plane, related to the Brownian web, but reflected and absorbed on a "barrier" which is the abscissa axis. In this work, we consider much more general barriers, construct an uncountable system of coalescing Brownian motions reflected and absorbed on these barriers, and the true self-repelling motion associated to it. The extension of the proofs of Tóth and Werner to this more general case is surprisingly difficult, especially when the barrier is irregular.
Paper Structure (19 sections, 39 theorems, 33 equations)

This paper contains 19 sections, 39 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Definitions and properties
  3. Forward lines
  4. Backward lines
  5. $(\lambda,\chi)$-true self-repelling motion
  6. Brownian motion and $(\lambda,\chi)$-RAB
  7. Brownian motion results
  8. $(\lambda,\chi)$-RAB results
  9. Forward lines: proof of Theorems \ref{['thm_cons_forward']} and \ref{['thm_forward_bis']}
  10. Lemma 8.1 of Toth_et_al1998
  11. Equation (8.22) of Toth_et_al1998
  12. Lemma 8.2 of Toth_et_al1998
  13. Equation (8.49) of Toth_et_al1998
  14. Backward lines: proof of Theorem \ref{['thm_cons_backward']}
  15. Markov property of the backward lines given $(\lambda,\chi)$
  16. ...and 4 more sections

Key Result

Theorem 6

The following properties hold. In addition, these properties characterize the law of $(\Lambda_{(x,h)})_{(x,h)\in \mathds{R}^2}$.

Theorems & Definitions (81)

  • Definition 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Theorem 7: Properties of the forward lines
  • Definition 8
  • Definition 9
  • Proposition 10
  • ...and 71 more