Hyperbolic State Space Decomposition for a Linear Stochastic Delay Equation

We consider the equation dX(t) = (∫- r⁰ X(s + t)dμ (s))dt + GdW(t) where $r > 0$, μ is a matrix-valued signed measure on $[ - r,0]$, W is an n-dimensional Wiener process and G is a fixed n × n matrix. Using the known decomposition C = S ⊕ 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.