Generation of Gaussian processes and linear chaos

Any stationary Gaussian process with a rational spectral density can be represented as the output of a linear infinite-dimensional Hamiltonian system in thermal equilibrium. The Hamiltonian system must have a continuous spectrum of the Lebesgue type. The authors show that on an extended phase space supporting invariant probability measures for the system, the Hamiltonian flow is hypercyclic, i.e., there is a vector generating a dense orbit. This is a well-known topological condition for chaos. In fact, the Hamiltonian flow is chaotic according to many standard definitions of the term