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The true self-repelling motion
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  • Published: July 1998

The true self-repelling motion

  • Bálint Tóth1 &
  • Wendelin Werner2 

Probability Theory and Related Fields volume 111, pages 375–452 (1998)Cite this article

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Abstract.

We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX t is equal to the gradient (in space) of L t (x), in a generalized sense, even though x↦L t (x) is not differentiable.

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary e-mail: balint@math-inst.hu, , , , , , HU

    Bálint Tóth

  2. Département de Mathématiques, Bât. 425, Université Paris-Sud, F-91405 Orsay Cedex, France e-mail: Wendelin.Werner@math.u-psud.fr, , , , , , FR

    Wendelin Werner

Authors
  1. Bálint Tóth
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  2. Wendelin Werner
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Received: 15 April 1997 / Revised version: 30 January 1998

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Tóth, B., Werner, W. The true self-repelling motion. Probab Theory Relat Fields 111, 375–452 (1998). https://doi.org/10.1007/s004400050172

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  • Issue Date: July 1998

  • DOI: https://doi.org/10.1007/s004400050172

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  • Mathematics Subject Classification (1991): 60G18
  • 60K35
  • 82C22
  • 82B41
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