A random attractor for a stochastic second order lattice system with random coupled coefficients (Q441946)

For the stochastic second order lattice system NEWLINE\[NEWLINE \frac{d^2u}{dt^2}+\alpha\frac{du}{dt} =-\lambda u+f(u)+g+A(\vartheta_t\omega)u+cu\,\frac{dW(t)}{dt}, NEWLINE\]NEWLINE where \(u=(u_k)_{k\in\mathbb Z}\), \(\alpha\), \(\lambda\) and \(c\) are constants, \(\alpha\) and \(\lambda\) positive, \(f(u)_k=f_k(u_k)\) with \(f_k\in C^1(\mathbb R)\), \(k\in\mathbb Z\), \(A\) is finite range, and~\(W\) is a Wiener process, conditions are given which allow to derive existence of a random set attractor. A similar result is claimed to hold for the additive noise case under slightly weaker conditions.