Mathematics – Probability
Scientific paper
2009-11-12
Mathematics
Probability
18 pages
Scientific paper
We study the asymptotic behavior of a self interacting one dimensional Brownian polymer first introduced by Durrett and Rogers. The polymer describes a stochastic process with a drift which is a certain average of its local time. We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence this enables us to partially prove a conjecture about the law of large numbers for the end-to-end displacement of the polymer formulated by Durrett and Rogers. Next we give upper and lower bounds for the variance of the process under the stationary measure, in terms of the qualitative infrared behavior of the interaction function. In particular we show that in the locally self-repelling case (when the process is essentially pushed by the negative gradient of its own local time) the process is super-diffusive.
Tarres Pierre
Toth Balint
Valkó Benedek
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