Hyperbolic State Space Decomposition for a Linear Stochastic Delay Equation

Abstract

We consider the equation $dX(t) = (\int_{ - r}^0 {X(s + t)d\mu (s))dt} + GdW(t)$ where $r > 0$, $\mu $ is a matrix-valued signed measure on $[ - r,0]$, W is an n-dimensional Wiener process and G is a fixed $n \times n$ matrix. Using the known decomposition $C = S \oplus U$ of $C = C[ - r,0]$ into a stable closed subspace S and a finite dimensional unstable subspace U for the analogous deterministic equation, we prove that the projection of the solution X onto S converges exponentially fast to a stationary Gaussian process. Also the speed of explosion of the projection onto U is exhibited.