Abstract:
In 1964, V. Arnold constructed an example of a nearly integrable
deterministic system exhibiting instabilities. In the 1970s, physicist B.
Chirikov coined the term for this phenomenon "Arnold diffusion", where
diffusion refers to stochastic nature of instability. One of the most
famous examples of stochastic instabilities for nearly integrable systems
is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were
discovered numerically by astronomer J. Wisdom. During the talk we describe
a class of nearly integrable deterministic systems, where we prove
stochastic diffusive behaviour. Namely, we show that distributions given by
deterministic evolution of certain random initial conditions weakly
converge to a diffusion process. This result is conceptually different from
known mathematical results, where existence of "diffusing orbits" is shown.
This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang,
and K. Zhang.