Open Access
March 2012 Diffusivity bounds for 1D Brownian polymers
Pierre Tarrès, Bálint Tóth, Benedek Valkó
Ann. Probab. 40(2): 695-713 (March 2012). DOI: 10.1214/10-AOP630


We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337–349]. 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 in Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337–349].

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.


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Pierre Tarrès. Bálint Tóth. Benedek Valkó. "Diffusivity bounds for 1D Brownian polymers." Ann. Probab. 40 (2) 695 - 713, March 2012.


Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1242.60105
MathSciNet: MR2952088
Digital Object Identifier: 10.1214/10-AOP630

Primary: 60K35 , 60K37 , 60K40
Secondary: 60F15 , 60G15 , 60J25 , 60J55

Keywords: asymptotic lower and upper bounds , Brownian polymers , Gaussian stationary measure , Local time , resolvent method , self-repelling random motion , strong theorems

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • March 2012
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