Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations (Q2471392)

The author proves an Oseledets-type theorem for a random functional differential equation \(\dot X (t) = A_t(\pi_t X)\). Here, \(E_{\gamma}\) denotes the space of continuous \({\mathbb {R}}^d\)-valued functions on \((-\infty,0]\) which are bounded by a multiple of \(\exp\{-\gamma s\}\) (\(\gamma > 0\)), \(\pi_t\) is the projection on \(E_{\gamma}\) defined by \(\pi_t(X)(s)=X(t+s)\,s \in (-\infty,0]\) and \(A_t\) is stationary with values in the space of bounded linear operators. The result is applied to a linear system with memory obtained from the linearized stochastic Navier-Stokes system on the 2D torus.