March 3, 2017
Location: Janskerkhof 15a (Utrecht), room 202 (morning) and 101 (afternoon)
Renormalization Group and Stochastic PDE’s I: Rough SPDE’s and Quantum Field Theory
I will introduce space-time white noise driven stochastic PDE’s, in particular the KPZ equation and the dynamical phi-4 model. Then I pose the problem of their renormalization and contrast it with the same problem in quantum field theory (QFT). Finally I introduce the Renormalization Group in the QFT setup and study it in the phi-4 model case in a simplified hierarchical approximation as a warmup for the dynamical case discussed in the subsequent lectures.
Phase transitions and uniqueness of g-measures
I will make a survey of the background as well as present the state of the art of research on g-measures. Doeblin and Fortet (1937) proved among other things that there is a unique stationary distribution for “chains of infinite connections” (finitely many states, but dependence on an infinite past) if the probability transition function has summable variations. Such stationary distributions were called g-measures by Michael Keane (1972), who introduced the modern theory. Since then there have been many contributors to the field of uniqueness results and also stronger ergodic theoretic properties of the g-measures (Ledrappier, Walters, Berbee, Hulse, Fernandez, Verbitskiy, and many others). There are also deep connections to rigorous statistical mechanics. I will also present some recent results by myself and my co-authors, Anders Johansson and Mark Pollicott.