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We study the asymptotic behavior of a self-interacting one-dimensionalBrownian polymer first introduced by Durrett and Rogers [Probab. Theory RelatedFields 92 (1992) 337--349]. The polymer describes a stochastic process with adrift which is a certain average of its local time. We show that a smeared outversion of the local time function as viewed from the actual position of theprocess is a Markov process in a suitably chosen function space, and that thisprocess has a Gaussian stationary measure. As a first consequence, this enablesus to partially prove a conjecture about the law of large numbers for theend-to-end displacement of the polymer formulated in Durrett and Rogers[Probab. Theory Related Fields 92 (1992) 337--349]. Next we give upper andlower bounds for the variance of the process under the stationary measure, interms of the qualitative infrared behavior of the interaction function. Inparticular, we show that in the locally self-repelling case (when the processis essentially pushed by the negative gradient of its own local time) theprocess is super-diffusive.

Diffusivity bounds for 1D Brownian polymers