Diffusivity bounds for 1D Brownian polymers
arXiv:0911.2356 · doi:10.1214/10-AOP630
Abstract
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.
Published in at http://dx.doi.org/10.1214/10-AOP630 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
References in corpus (2)
Cited by in corpus (8)
- Anomalous fluctuations for a perturbed Hamiltonian system with exponential interactions
- Self-repelling diffusions via an infinite dimensional approach
- Diffusive limits for "true" (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d >= 3
- Self-repelling diffusions on a Riemannian manifold
- Inverting the Ray-Knight identity on the line
- Diffusion and superdiffusion in lattice models of colliding particles with stored momentum
- -superdiffusivity for a Brownian particle in the curl of the 2d GFF
- Exit-problem for a class of non-Markov processes with path dependency