The asymptotic behavior of an infinite system of connected oscillators (Q1896942)

Suppose on a probability space \((\Omega, \sigma, P)\) an increasing flow \((F_t)_{t \geq 0}\) of \(\sigma\)-algebras is given. We consider the infinite-dimensional diffusion process \(\xi(t) = (\xi (t,x), z \in \mathbb{Z}^\nu)\), defined by the system of Itô equations \[ d \xi (t,z) = w_z dt + dW(t,x), \tag{2} \] where \(W(t,z)\), \(t \geq 0\), are standard Wiener processes coordinated with \(F_t\) and independent for distinct \(z \in \mathbb{Z}^\nu\), \(w_z \in \mathbb{R}\) are ``fundamental frequencies'' with the initial condition \(\xi (0,z) = u_z \in \mathbb{R}\). We consider the case of weak interaction of processes (2). However, our results are comparable with numerical experiments and enable us to calculate asymptotic relations of mean phases for different oscillators with sufficient accuracy.