Common frequently hypercyclic random vectors

We study common frequently hypercyclic vectors for countable families of weighted backward shifts acting on ${\mathrm{\ell }}_p$ spaces, $1\le p<\mathrm{\infty }$. Using probabilistic techniques, we develop a general existence criterion, complemented by a non-existence result. These insights are then applied to the specific setting of countable families of polynomials of weighted backward shifts, providing conditions under which they share a common frequently hypercyclic vector.