  A theorem of Godefroy and Shapiro states that non-trivial convolution
operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic.
Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently
hypercyclic functions of exponential growth. On the other hand, in the infinite
dimensional setting, the Godefroy-Shapiro theorem has been extended to several
spaces of entire functions defined on Banach spaces. We prove that on all these
spaces, non-trivial convolution operators are strongly mixing with respect to a
gaussian probability measure of full support. For the proof we combine the
results previously mentioned and we use techniques recently developed by Bayart
and Matheron. We also obtain the existence of frequently hypercyclic entire
functions of exponential growth.
