The true self-repelling motion

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 (Xt, 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 Lt(x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: -dXt is equal to the gradient (in space) of Lt(x), in a generalized sense, even though x → Lt(x) is not differentiable.