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Probability Theory and Related Fields

, Volume 111, Issue 3, pp 375–452 | Cite as

The true self-repelling motion

  • Bálint Tóth
  • Wendelin Werner
Article

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 xL t (x) is not differentiable.

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Bálint Tóth
    • 1
  • Wendelin Werner
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary e-mail: balint@math-inst.huHU
  2. 2.Département de Mathématiques, Bât. 425, Université Paris-Sud, F-91405 Orsay Cedex, France e-mail: Wendelin.Werner@math.u-psud.frFR

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