Annals of Probability

Diffusivity bounds for 1D Brownian polymers

Pierre Tarrès, Bálint Tóth, and Benedek Valkó

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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.

Article information

Ann. Probab., Volume 40, Number 2 (2012), 695-713.

First available in Project Euclid: 26 March 2012

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60K40: Other physical applications of random processes
Secondary: 60F15: Strong theorems 60G15: Gaussian processes 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals

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


Tarrès, Pierre; Tóth, Bálint; Valkó, Benedek. Diffusivity bounds for 1D Brownian polymers. Ann. Probab. 40 (2012), no. 2, 695--713. doi:10.1214/10-AOP630.

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