# The true self-repelling motion

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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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