Brownian Polymers

Brownian Polymers

Mon, 14/02/2011
15:45 		Pierre Tarres Stochastic Analysis Seminar Add to calendar Eagle House
Brownian Polymers
We consider a process $ X_t\in\R^d $, $ t\ge0 $, introduced by Durrett and Rogers in 1992 in order to model the shape of a growing polymer; it undergoes a drift which depends on its past trajectory, and a Brownian increment. Our work concerns two conjectures by these authors (1992), concerning repulsive interaction functions $ f $ in dimension $ 1 $ ($ \forall x\in\R $, $ xf(x)\ge0 $). We showed the first one with T. Mountford (AIHP, 2008, AIHP Prize 2009), for certain functions $ f $ with heavy tails, leading to transience to $ +\infty $ or $ -\infty $ with probability $ 1/2 $. We partially proved the second one with B. Tóth and B. Valkó (to appear in Ann. Prob. 2011), for rapidly decreasing functions $ f $, through a study of the local time environment viewed from the particule: we explicitly display an associated invariant measure, which enables us to prove under certain initial conditions that $ X_t/t\to_{t\to\infty}0 $ a.s., that the process is at least diffusive asymptotically and superdiffusive under certain assumptions.