Lai-Sang Young (Courant Institute and IAS, Princeton)
Title:
Observable events and typical trajectories in dynamical systems
Abstract:
I will present ideas related to "typical solutions" for finite and infinite
dimensional dynamical systems, deterministic and stochastic. In finite
dimensions, one often equates observable events with positive Lebesgue
measure sets, and view invariant measures that reflect large-time
behaviors of positive Lebesgue measure sets of initial conditions as
physically relevant. Accepting these ideas, there is a simple and very
nice picture that one might hope to be true. Reality is messier,
unfortunately, at least for deterministic systems. I will argue that
the addition of a small amount of random noise will improve the
situation. As for infinite dimensional systems, such as those defined
by semi-flows generated by evolutionary PDEs, a different notion of
observability is needed. I will finish with some results that suggest
a vaible notion of "typical solutions" that connects with the notion of
observability in finite dimensions.