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Lyapunov exponents and stationary solutions for affine stochastic delay equations. (English) Zbl 0704.60062
The authors consider stochastic functional DEs of the type of affine delay equations $$ (*)\quad dX(t)=HX\sb tdt+dM(t),\quad X\sb 0=y,\quad t\ge 0, $$ where the driving force M is a stochastic process with stationary increments or locally integrable deterministic, y and $X\sb t$ with $X\sb t(u)=X(t+u)$, $t\ge 0$, are ${\bbfR}\sp n$-valued Lebesgue integrable functions on a fixed interval $J=[-r,0]$, and H is the bounded linear operator defined by $Hy=\int\sb{J}dm(u)y(u)$ with an $n\times n$- matrix valued function m of bounded variation. The Lyapunov spectrum of the associated homogeneous DE is used to decompose the state space into finite-dimensional and finite- codimensional subspaces and to investigate the (asymptotics and) Lyapunov exponents of the solution X(t) via projections onto these subspaces. If the associated homogeneous DE has no vanishing Lyapunov exponents and the driving force M has stationary increments, there exists a unique stationary solution X(t). For this situation a description of the almost sure Lyapunov spectrum and results on the p th moment Lyapunov exponents of (*) are given.
Reviewer: V.Wihstutz

60H25Random operators and equations
34K25Asymptotic theory of functional-differential equations
60G10Stationary processes
93E20Optimal stochastic control (systems)
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