Stability of stationary solutions. (Q1432124)

The author considers the equation \( \varepsilon x''(t)+x'(t)=Ax(t)+\xi(t),\) where \(x\) and \(\xi\) are Banach space valued functions on the real line, \(A\) is a bounded linear operator and \(\xi\) is a stationary process that is smooth in time. He announces the result that a unique stationary solution \(x_\varepsilon\) to this equation exists with \(x_\varepsilon\to x_0\) uniformly for \(\varepsilon\to 0\), where \(x_0\) is the unique stationary solution for the equation with \(\varepsilon=0\). In the second part a boundary-value problem depending on a parameter is briefly considered.