Abstract:

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.

Lecture 1:

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.

Lecture 2:

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)

Lecture 3:

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, Yaglom-reversibility.

Lecture 4:

Kipnis-Varadhan type theorems for additive functionals of Markov processes -- non-reversible cases. (Survey)

Lecture 5:

Application of the Kipnis-Varadhan type of approach to (a) myopic self-repelling walks in 3d; (b) some Durrett-Rogers processes. Outlook.