  What kind of dynamics do we observe in general on the circle? It depends
somehow on the interpretation of "in general". Everything is quite well
understood in the topological (Baire) setting, but what about the probabilistic
sense? The main problem is that on an infinite dimensional group there is no
analogue of the Lebesgue measure, in a strict sense. There are however some
analogues, quite natural and easy to define: the Malliavin-Shavgulidze measures
provide an example and constitute the main character of this text. The first
results show that there is no actual disagreement of general features of the
dynamics in the topological and probabilistic frames: it is the realm of
hyperbolicity! The most interesting questions remain however unanswered... This
work, coming out from the author's Ph.D. thesis, constitutes an opportunity to
review interesting results in mathematical topics that could interact more
often: stochastic processes and one-dimensional dynamics. After an introductory
overview, the following three chapters are a pedagogical summary of classical
results about measure theory on topological groups, Brownian Motion, theory of
circle diffeomorphisms. Then we present the construction of the
Malliavin-Shavgulidze measures on the space of interval and circle $C^1$
diffeomorphisms, and discuss their key property of quasi-invariance. The last
chapter is devoted to the study of dynamical features of a random
Malliavin-Shavgulidze diffeomorphism.
