Bálint Tóth, Budapest : "Scaling limits for self-interacting random walks and diffusions"
I will discuss the long time asymptotics of random walks and
diffusions with long memory due to path-wise self-interactions defined
in terms of the (gradients of) their own local times. These include
so-called myopic self-repelling walks, some Durrett-Rogers polymers
and other related processes.
General class of models discussed. Survey of historical background.
Myopic (or 'true') self-repelling random walks on Z^d. Durrett-Rogers
polymer processes. Main problems and conjectures.
Limit theorem with time-to-the-2/3-scaling in 1d. Ray-Knight-type
approach. Some other related limit theorems.
Construction of the continuous time "true self-repelling motion" - by
use of so-called Brownian Web. (part 1)
True self-repelling motion continued (and hopefully finished by half-time).
The environment as seen by the random walker: random walk in a
particular dynamical random environment. Stationarity, ergodicity,
Kipnis-Varadhan type theorems for additive functionals of Markov
processes -- non-reversible cases. (Survey)
Application of the Kipnis-Varadhan type of approach to (a) myopic
self-repelling walks in 3d; (b) some Durrett-Rogers processes.