Some results concerning the existence of almost surely frequently hypercyclic random
vectors have been proved in the literature for certain chaotic weighted shifts. This is
of interest for at least two reasons. It is usually difficult to find explicit (frequently)
hypercyclic vectors, and random vectors have a probability distribution whose ergodic
properties can be studied. The first objective of the thesis is to extend the previously
known results. In particular, we prove that every chaotic weighted shift on very
general sequence spaces and every operator satisfying the Frequent Hypercyclicity
Criterion admits an almost surely frequently hypercyclic random vector.
We also investigate the case of semigroups. The desired random vector is con structed using a stochastic integral. Although our general result requires that this
integral is well-defined, we can apply it to the translation semigroups on the space of
entire functions.
The second part of the thesis deals with the rate of growth of frequently hypercyclic
functions. We present two methods. Recently, a probabilistic approach provided a
quasi-optimal rate of growth for the differentiation operator and the Taylor shift.
Based on these results and the first part of the thesis, we obtain a general criterion
for chaotic weighted shifts. The rate of growth is expressed as a function depending
only on the weights, multiplied by some logarithmic factor. We give several examples
of shifts defined on the space of entire functions or the space of holomorphic functions
on the unit disk, recovering previous results and finding new ones. We also consider
the differentiation operators on the space of harmonic functions on the plane and
weighted shifts on Köthe sequence spaces. The possible optimality of the growth is
also discussed.
On spaces of holomorphic functions, we can also ask whether the growth holds
outside some small, but possibly unbounded, set. We give results in this direction,
which are stated for general random complex series. This second approach seems to
be new in linear dynamics. In particular, we prove that for any chaotic weighted shift,
the growth sought by the previous method does hold outside such a set
