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%\topmatter
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%\topmatter


%\title
{\aa
\centerline{{  \a IV. ERGODIC THEORY OF REGULAR LINEAR SFDE's    }}
%\line{}
%\centerline{{\a CLASSIFICATION OF SFDE'S 
 %}} 
\line{}
\line{}
 

\vbox to 1.0truein{}
 

{\aa
\centerline{{  \a Geilo, Norway   }}
\line{}
%\vbox to 1.0truein{}
  \centerline{{\a  Thursday, August 1, 1996}}
\line{}
\centerline{{\a 14:00-14:50}}
%\author
\vbox to 0.5truein{}
{\a
\centerline{ Salah-Eldin A. Mohammed  }}
\line{}
{\d
\centerline{Southern Illinois University}
\line{}
\centerline{Carbondale, \ IL \ 62901--4408 \ USA }
\line{}
\centerline{Web page: \bf{http://salah.math.siu.edu}}}

%\vbox to 0.5truein{}
 
 

%\rightheadtext{I}
  
%\leftheadtext{I}




\newpage

%\baselineskip=36truept
%\baselineskip=24truept
%\baselineskip=30truept



\baselineskip=18truept






{\a
 \centerline{IV.  ERGODIC THEORY OF LINEAR SFDE's }
}





\noindent{\a 1.  Plan}

{\d

Use state space $M_2$. 
%$|\cdot|$:= Euclidean norm on $\br^d$.
For regular linear sfde's (VIII), (IX), consider the following themes:


\item{I)} Existence of a ``perfect" cocycle on $M_2$ that is a modification 
of the trajectory field $(x(t), x_t) \in M_2$.

\item{II)} Existence of almost sure Lyapunov exponents 
%
$$ \lim_{t \to \infty} \frac{1}{t}\, \log \| (x(t),x_t)\|_{M_2}$$
%
\item{}
The multiplicative ergodic theorem and  {\ii hyperbolicity\/} of  the cocycle.


\item{III)}{\ii The Stable Manifold Theorem}, (viz. ``random saddles") for hyperbolic systems. 

}

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\newpage
\document
\baselineskip=24truept

 
\newpage

                       

\newpage
\noindent{\a 2.  Regular Linear Systems. White Noise}

{\d
   Linear  sfde's on $\bold R^d$ driven by $m$-dimensional Brownian motion  $W:= (W_1,\cdots,W_m)$, with smooth coefficients. 
\noindent  
%
$$
     \left. \aligned
     dx(t) &= H(x(t-d_1), \cdots, x(t-d_N), x(t), x_t)dt \\ 
     &\qquad \qquad +\sum_{i=1}^{m}g_i x(t)\,dW_i(t), \quad  t > 0    \\
    (x(0), x_0)&=(v,\eta) \in M_2:= \R^d \times L^2 ([-r,0],\R^d)
     \endaligned
     \right \}
                                        \tag{VIII}
$$
%

 (VIII) is defined on 

$(\O, \F, (\F_t)_{t \in \R}, P)=$  canonical complete filtered Wiener space.

 $\O:=$  space of all continuous paths $\w: \br \to \br^m$, $\w(0) = 0$, in Euclidean space $\br^m$, with  compact open topology; 

$\F:=$ completed Borel $\sigma$-field of $\O$; 

$\F_t:=$ completed sub-$\sigma$-field of $\F$ generated by the evaluations
$\o \to \o(u), \,\,$  $u \leq t, \quad t \in \R$.

 $P:=$  Wiener measure  on $\O$.

 $dW_i(t)=$ It\^o stochastic differentials.


Several finite delays $0 < d_1 < d_2 < \dots < d_N \leq r$ in
 drift term; {\ii no delays in diffusion coefficient}.

$H: ({\R}^d) ^{N + 1} \times  L^2 ([-r,0], {\R}^d)\to \R^d$
is a fixed continuous linear map, $g_i$, $i = 1, 2, \dots, m$, fixed (deterministic) $d \times d$-matrices.

Recall regularity theorem:
%\newpage
\medskip



\noindent{\a  Theorem III.4.}([Mo], Stochastics, 1990])  

{\sl

 (VIII) is regular with respect to the state space
$ M_2 = {\R}^d \times \bold L ^2 ([-r,0], {\R}^d)$.  There is a  
measurable version $X: {\R}^+\times M_2 \times \Omega  \to M_2$ of the trajectory field $\{ (x(t),
x_t) : t \in {\R}^+$, $(x (0), x_0) = (v, \eta) \in M_2 \}$ of (VIII) with
the following properties:

\itemitem{(i)}  For each $(v, \eta) \in M_2$ and $t \in {\R}^+$, $X(t, (v, \eta), \cdot)
= (x (t), x_t)$   a\.s., is ${\Cal F}_t$-measurable and belongs to
$ L^2(\Omega, M_2; P)$.

\itemitem{(ii)}  There exists $\Omega _0 \in \F$ of full measure such that, for all $\omega \in \Omega _0$, the map
$X(\cdot,\cdot, \omega ): {\R}^+ \times M_2 \to M_2$ is continuous.

\itemitem{(iii)}  For each $t \in {\R}^+$ and every $\omega \in
\Omega _0$, the map $X(t,\cdot, \omega ) : M_2 \to M_2$ is continuous
linear; for each $\omega \in \Omega _0$, the map ${\R}^+ \ni
t \mapsto X(t,\cdot, \omega) \in L(M_2)$ is measurable and locally
bounded in the uniform operator norm on $L(M_2)$. The map $[r, \infty )\ni t \mapsto X(t,\cdot, \omega) \in L(M_2)$ is continuous for all $\omega \in \Omega _0$.

\itemitem{(iv)}  For each $t \geq r$ and all $\omega \in \Omega _0$, the map $$X(t,\cdot, \omega ): M_2 \to M_2$$ 
\itemitem{} 
is compact.
}


 
\bigskip
\medskip
                        
{\ii Compactness of semi-flow for $t \geq r$ will be used below to define hyperbolicity
for (VIII) and the associated exponential dichotomies.\/} 


}





%\newpage
\bigskip
\noindent{\a Lyapunov Exponents.  Hyperbolicity}
 
\medskip
 {\d

 Version $X$ of the flow constructed in Theorem III.4 is a multiplicative $L(M_2)$-valued linear cocycle over the canonical Brownian shift $\theta : {\R} \times \Omega \to \Omega$ on Wiener space:
%
$$
     \theta (t, \omega) (u) := \omega (t + u) - \omega (t), \quad u,
t \in {\R}, \quad \omega \in \Omega.
$$
%
\noindent
Indeed we have

\bigskip
\noindent
\noindent{\a  Theorem IV.1}([M], 1990) 

{\sl

There is an ${\Cal F}$-measurable set $\hat \Omega$ of full
$P$-measure such that $\theta (t, \cdot) (\hat \Omega) \subseteq \hat
\Omega$ for all $t \geq 0$ and
%
$$
     X(t_2,\cdot, \theta (t_1, \omega))\circ X(t_1,\cdot, \omega) =
X(t_1 + t_2,\cdot, \omega) 
$$
%
\noindent
for all $\omega \in \hat \Omega$ and $t_1$, $t_2 \geq 0$.
} 

\newpage
{\d
 \medskip
\centerline{{\ii The Cocycle Property}}

\vbox to 4.0truein{}



}





\newpage
\noindent{\a Proof of Theorem IV.1.} (Sketch)

{\d
  
For simplicity consider the case of a single delay $d_1$; i.e. $N=1$.    
}

%\newpage
\noindent
{\a{\ii First step.\/}}  

{\d
Approximate the Brownian motion $W$ in (VIII)
by smooth adapted processes $\{ W^k \}^\infty _{k = 1}$:
%
$$
     W^k(t) := k \int ^t _{t - (1/k)} W(u) \,du -  k \int ^0 _{- (1/k)} W(u) \,du, \quad t \geq 0,\,\, k \geq 1.\tag{1}
$$
%
\noindent
{\ii Exercise:} Check that each $W^k$ is a {\ii helix\/} (i.e. has stationary increments):
%
$$W^k (t_1+t_2,\o) - W^k (t_1,\o) = W^k (t_2, \theta (t_1,\o)), \quad t_1,t_2 \in \R, \,\, \o \in \O.  \tag{2}
$$
%
Let $X^k : {\R}^+  \times M_2\times \O \to M_2$ be the
stochastic (semi)flow of the random  fde's:
%
$$
     \left. \aligned
     dx^k(t) &= H(x^k(t-d_1), x^k (t), x^k_t)dt \\ 
     & \quad +\sum_{i=1}^{m}g_i x(t) (W_i^k)'(t)\,dt - \frac{1}{2} \sum_{i=1}^m g_i^2 x^k (t) \,dt  \quad  t > 0    \\
    (x^k(0), x^k_0)&=(v,\eta) \in M_2:= \R^d \times L^2 ([-r,0],\R^d)
     \endaligned
     \right \}
                                        \tag{VIII-k}
$$



     If $X : {\R}^+  \times M_2 \times \Omega \to M_2$ is the flow
of (VIII) constructed in Theorem III.4, then
%
$$
     \lim _{k \to \infty} \, \sup _{0 \leq t \leq T} \| X^k (t,\cdot,\omega) - X(t,\cdot, \omega) \| _{L(M_2)} = 0 \tag{3}
$$
%
\noindent
for every $0 < T < \infty$ and all $\omega$ in a
Borel set $\hat \Omega$ of full Wiener measure which is invariant
under $\theta (t, \cdot)$ for all $t \geq 0$ ([Mo], Stochastics, 1990). This convergence may be proved using the following stochastic variational method:

    Let $\phi :{\R}^+   \times \Omega \to {\R}^{d\times d}$ be the $d \times d$-matrix-valued solution of the linear It\^o  sode (without delay):
%
$$
 \left.     \aligned
     d \phi (t) &= \sum_{i=1}^m g_i \phi (t) \; d W_i (t) \qquad t > 0 \\
     \phi ( 0,\o) &= I \in {\R}^{d \times d} \qquad \hbox{ a.a.$\,\,\w \,\,$}
     \endaligned \right \} \tag{4} 
$$
%
\noindent
Denote by  $\phi^k :{\R}^+   \times \Omega \to {\R}^{d\times d},\,\, k \geq 1$, the $d\times d$-matrix solution of the random family of linear ode's:
%
$$
 \left.     \aligned
     d \phi^k (t) &= \sum_{i=1}^m g_i \phi^k (t)  (W_i^k)' (t) - \frac{1}{2} \sum_{i=1}^m g_i^2 \phi^k (t) \,dt   \qquad t > 0 \\
     \phi^k ( 0,\cdot) &= I \in {\R}^{d \times d}.  
     \endaligned \right \} \tag{4'} 
$$
%
\noindent
 Let $\hat\O $ be the sure event of all $\o \in \O$ such that
% 
$$\phi (t,\o):= \lim_{k \to \infty} \phi^k (t,\o)  \tag{5}$$
%
exists uniformly for $t$ in compact subsets of $\R^+$. Each $\phi^k$ is an
$\R^{d\times d}$-valued {\ii cocycle over $\theta$}, viz.
%
$$
\phi^k (t_1 + t_2,\omega) = \phi^k (t_2, \theta (t_1, \omega)) \phi^k (t_1,\omega) \tag{6}
$$ 
%
for all $t_1, t_2 \in {\R}^+$ and $\omega \in \Omega $.
From the definition of $\hat\O$ and passing to the limit in (6) as $k \to \infty$,
conclude that $\{\phi (t,\w)  :t > 0, \,\,\w \in \Omega\}$,  is  an $\R^{d\times d}$-valued {\ii perfect \/} cocycle over $\theta$, viz. 
 \noindent
 
\itemitem{(i)}  $P(\hat\Omega) = 1$;

\itemitem{(ii)}  $\theta (t, \cdot) (\hat\Omega ) \subseteq \hat\Omega $
for all $t \geq 0$;

\itemitem{(iii)}  $\phi (t_1 + t_2,\omega) = \phi (t_2, \theta
(t_1, \omega)) \phi (t_1, \omega)$ for all 
\itemitem{} $t_1, t_2 \in {\R}^+$ and {\ii every} $\omega \in \hat\Omega $;

\itemitem{(iv)}  $\phi (\cdot, \omega)$ is continuous for every
$\omega \in \hat\Omega $.
 
    
 Alternatively use the  perfection theorem in ([M-S], AIHP, 1996, Theorem 3.1, p. 79-82) for crude cocycles with values in a metrizable second countable topological group. Observe that $\phi (t,\o) \in GL (\R^d)$.
   

     Define $\hat H : {\R}^+ \times \R^d \times M_2 \times \O \to {\R}^d$ by
%
$$
\align  
\hat H (t,&v_1,v, \eta,\o)\\ 
&:=  \phi (t,\o)^{-1} [ H ( \phi_t (\cdot,\o)(-d_1,v_1) , \phi (t,\o)(v),\phi _t ( \cdot,\o) \circ (id_J, \eta))] \tag{7}
\endalign
$$
%
for $ \w \in \Omega,t \geq 0,\,\,v,v_1 \in \R^d,\,\, \eta \in L^2([-r,0],\R^d)$,
%
\noindent
where
%
$$
     \phi_t ( \cdot,\w) (s, v) = \cases \phi ( t + s,\o)(v) &\qquad t + s \geq 0 \\
     & \\
     v &\qquad -r \leq t + s < 0 \endcases
$$
%
\noindent
and
%
$$(id_J, \eta) (s) = (s, \eta(s)),\,\, \quad s \in J.$$
%
Define  $\hat H^k : {\R}^+  \times \R^d \times M_2 \times \O \to {\R}^d$ by 
a relation similar to (7) with $\phi$ replaced by $\phi^k$.
    Then the random fde's
%
$$
     \aligned
     y'(t) &= \hat H(t,y(t-d_1), y(t), y_t,\o) \qquad t > 0 \\
    (y(0), y_0 )&= (v,\eta) \in M_2
     \endaligned \qquad \biggr \} \tag{8}
$$
%
\medskip
$$
     \aligned
     {y^k}'(t) &= \hat H^k (t,y^k (t-d_1), y^k (t), y^k_t,\o) \qquad t > 0 \\
    (y^k(0), y^k_0 )&= (v,\eta) \in M_2
     \endaligned \qquad \biggr \} \tag{9}
$$
\noindent
have  unique {\ii non-explosive} solutions $$y,\, y^k : [ - r,\infty)\times \O \to {\R} ^d \, \,$$([Mo], Stochastics, 1990, pp. 93-98). It\^o's formula implies that  
%
$$
     X(t,v, \eta,\o) = (\phi (t,\o)(y(t,\o)),\phi_t ( \cdot,\o) \circ (id_J, y_t)) \tag{10}
$$
%
\noindent
  The chain rule gives a similar relation for
$X^k$ with $\phi$ replaced  by $\phi^k$ ({\ii Exercise}; [Mo], Stochastics, 1990, pp. 96-97).

Get the convergence
%
$$
\lim_{k \to \infty} |\hat H^k (t,v_1, v, \eta,\o)- \hat H (t,v_1, v, \eta,\o)|=0 \tag{11}
$$
uniformly for $(t,v_1,v,\eta)$ in bounded sets of $\R^+ \times \R^d \times M_2$. Use Gronwall's lemma and (11) to deduce (3).



%\qd
%\newpage
\noindent
{\ii Second step.\/}  

Fix $\omega \in \hat \Omega$
and use uniqueness of solutions to the approximating equation (VIII-k) and the helix property (2) of $W^k$ to obtain the
cocycle property for $(X^k, \theta)$:  
%
$$
     X^k (t_2,\cdot, \theta (t_1, \omega))\circ X^k (t_1,\cdot, \omega) =
X^k (t_1 + t_2,\cdot, \omega) 
$$
%
\noindent
for all $\omega \in  \hat\Omega$ and $t_1$, $t_2 \geq 0$, $k \geq 1$.  

\noindent
{\ii Third step.\/}  

Pass to limit as $k \to\infty$ in the above identity and use the convergence (3) {\ii in operator norm} to get the perfect cocycle property for $X$.
}   \qd

\newpage


{\d

%\newpage
\bigskip
\medskip

    The a\.s\. Lyapunov exponents
%
$$
     \lim _{t \to \infty} \frac {1}{t} \log \| X(t, (v(\omega), \eta(\omega)), \omega) \| _{M_2},  
$$
%
\noindent
(for a\.a\.  $\omega \in \Omega,\,\, (v, \eta) \in  L^2 (\Omega, M_2)$)   of the system (VIII) are characterized by the following ``spectral theorem".
Each $\theta (t,\cdot)$ is ergodic and preserves Wiener measure $P$.
The proof of Theorem IV.2 below uses compactness of $X(t,\cdot, \omega): M_2 \to M_2$, $t \geq r$, together with an infinite-dimensional version of
Oseledec's multiplicative ergodic theorem due to Ruelle (1982).
 }


%\newpage
\bigskip
%\noindent
 \noindent{\a  Theorem IV.2.} ([Mo], Stochastics, 1990) 

{\sl
Let $X : {\R}^+ \times M_2 \times \Omega  \to M_2$ be the flow of
(VIII) given in Theorem III.4.  Then there exist

\itemitem{(a)}  an $\Cal F$-measurable set $\Omega ^* \subseteq \Omega$ such that
$P(\Omega ^*) = 1$ and 
\newline $\theta (t, \cdot) (\Omega ^*) \subseteq
\Omega ^*$ for all $t \geq 0$,

\itemitem{(b)}  a fixed (non-random) sequence of real numbers $\{
\lambda _i \}^\infty _{i = 1}$, and

\itemitem{(c)}  a random family $\{ E_i (\omega) : i \geq 1, \omega
\in \Omega ^* \}$ of (closed) finite-codimensional subspaces of
$M_2$, with the following properties:

\itemitem{}  (i)  If the {\bf Lyapunov spectrum} $\{ \lambda _i
\}^\infty _{i = 1}$ is infinite, then $\lambda _{i + 1} < \lambda _i$
for all $i \geq 1$ and $\lim \limits _{i \to \infty} \lambda _i = -
\infty$; otherwise there is a fixed (non-random) integer $N \geq 1$
such that $\lambda _N = - \infty < \lambda _{N - 1} < \dots < \lambda
_2 < \lambda _1$; \newline
(ii)  each map $\omega \mapsto E_i (\omega)$, $i \geq 1$, is ${\Cal
F}$-measurable into the Grassmannian of $M_2$; \newline
(iii)  $E_{i + 1} (\omega) \subset E_i (\omega) \subset \dots \subset E_2 (\omega) \subset E_1 (\omega) = M_2$, $i
\geq 1,\, \omega \in \Omega ^*$; \newline
(iv)  for each $i \geq 1$, codim $E_i (\omega)$ is fixed independently
of $\omega \in \Omega ^*$;\newline
(v)  for each $\omega \in \Omega ^*$ and $(v, \eta) \in E_i (\omega)
\backslash E_{i + 1} (\omega)$,
%
$$
     \lim _{t \to \infty} \frac {1}{t} \log \| X(t,(v,
\eta), \omega ) \| _{M_2} = \lambda _i, \, i \geq 1;
$$
%
\itemitem{}  (vi)  {\bf Top Exponent}:
%
$$
     \lambda _1= \lim _{t \to \infty} \frac {1}{t} \log \| X(t,\cdot,
\omega) \| _{L(M_2)} \quad \hbox{ for all } \omega \in \Omega^*;
$$
%
\itemitem{}  (vii) {\bf Invariance}:
%
$$
     X(t,\cdot,\omega) (E_i (\omega)) \subseteq E_i (\theta (t, \w))
$$
%
for all $\,\omega \in \Omega ^*,\,\, t \geq 0,\,\, i \geq 1$.

} 


\newpage
\bigskip
%\medskip
%\newpage
\centerline{{\ii Spectral Theorem }}




\newpage





{\d
    
 Proof of Theorem IV.2 is based on Ruelle's discrete version of
Oseledec's multiplicative ergodic theorem in Hilbert space ([Ru], Ann. of Math. 1982, Theorem (1.1), p\. 248 and Corollary (2.2), p\. 253):
}

\medskip
\noindent{\a Theorem IV.3 } ([Ru], 1982)
 
{\sl  

Let $(\O, \F, P)$ be a probability space and $\tau :\O \to \O$ a $P$-preserving transformation. Assume that $H$ is a separable Hilbert space and $T: \O \to L(H)$ a measurable map (w.r.t. the Borel field on the space of all bounded linear operators $L(H)$).
Suppose that $T(\o)$ is compact for almost all $\o \in \O$, and 
$ E \log^+ \|T(\cdot)\| < \infty.$  
Define the family of linear operators $\{ T^n (\o): \o \in \O, \,\, n \geq 1\}$ by
%
$$ T^n (\o):= T(\tau^{n-1} (\o))\circ \cdots T(\tau (\o))\circ T(\o)$$
%
for $\o \in \O, \,\, n \geq 1$. 

Then there is a set $\O_0 \in \F$ of full $P$-measure such 
that \newline $\tau (\O_0) \subseteq \O_0$, and for  each $\o \in \O_0$, the limit
%
$$ \lim_{n \to \infty} [T^n (\o)^* \circ T^n (\o)]^{1/(2n)} := \Lambda (\o)$$
%
exists in the uniform operator norm and is a positive compact self-adjoint operator on $H$. Furthermore
each $\Lambda (\o)$ has a discrete spectrum
%
$$ e^{\mu_1 (\o)} >  e^{\mu_2 (\o)}  > e^{\mu_3 (\o)} > e^{\mu_4 (\o)} > \cdots$$
%
where the $\mu_i$'s are distinct. If  $\{\mu_i \}_{i=1}^{\infty}$ is infinite, then $\mu_i \downarrow -\infty$; otherwise they terminate at $\mu_{N(\o)}= -\infty$. If $\mu_i (\o) > -\infty$, then $e^{\mu_i (\o)}$ has finite multiplicity $m_i (\o)$ and finite-dimensional eigen-space $F_i (\o)$, with $m_i (\o):= \hbox{dim} F_i (\o)$. Define 
%
$$  E_1 (\o):= M_2,\quad E_i (\o):= \bigl [\displaystyle \oplus_{j=1}^{i-1} F_j (\o) \bigr]^{\perp}, \quad
E_{\infty} (\o):= \hbox{ker}\,\, \Lambda (\o).$$
%
Then
%
$$ 
E_{\infty} (\o) \subset \cdots \subset E_{i+1}(\o)\subset E_{i} (\o) \cdots \subset E_2 (\o) \subset E_1 (\o) =H
$$
% 
and 
%
$$ 
\lim_{n \to \infty} \frac{1}{n} \log \|T^n (\o) x\|_H =\cases  \mu_i (\o), \quad \hbox{if} \quad x \in E_i (\o) \backslash E_{i+1} (\o)\\
 -\infty \qquad \hbox{if} \quad x \in \hbox{ker}\,\, \Lambda (\o).
\endcases
$$
%


}

%\newpage
\noindent
{\a Proof. }

{\d  [Ru], Ann. of Math., 1982, pp. 248-254. \qd}

\bigskip

{\d

The following
``perfect" version of Kingman's subadditive ergodic theorem is also used
to construct the shift invariant set $\Omega ^*$ appearing in Theorem
IV.2 above.

 }
\bigskip


 \newpage


 \noindent{\a  Theorem IV.4}([M, 1990])(``Perfect" Subadditive Ergodic Theorem) 

{\sl
Let $f : {\R}^+ \times \Omega \to {\R} \cup \{ - \infty \}$
be a measurable process on the complete probability space $(\Omega,
{\Cal F}, P)$ such that
\item{(i)}  $E \sup \limits _{0 \leq u \leq 1} f^+(u, \cdot) <
\infty$, $E \sup \limits _{0 \leq u \leq 1} f^+ (1 - u, \theta (u,
\cdot)) < \infty$;

\item{(ii)}  $f (t _1 + t_2, \omega) \leq f (t_1, \omega) + f(t_2,
\theta (t_1, \omega))$ for all $t_1, t_2 \geq 0$ and {\bf every} $
\omega \in \Omega$.

\noindent
Then there exist a set $\hat {\hat \Omega} \in {\Cal F}$ and a
measurable $\tilde f : \Omega \to {\R} \cup \{ - \infty \}$ with the properties:

\itemitem{(a)}  $P (\hat {\hat \Omega}) = 1,\,\, \theta (t, \cdot) (\hat {
\hat \Omega}) \subseteq \hat {\hat \Omega}$ for all $t \geq 0$;

\itemitem{(b)}  $\tilde f (\omega) = \tilde f (\theta (t, \omega))$ for all
$\omega \in \hat {\hat \Omega}$ and all $t \geq 0$;

\itemitem{(c)}  $\tilde f^+ \in \bold L^1 (\Omega, {\R}; P)$;

\itemitem{(d)}  $\lim \limits _{t \to \infty} (1/t) f(t, \omega) =
\tilde f (\omega)$ for every $\omega \in \hat {\hat \Omega}$.

\noindent
If $\theta$ is ergodic, then there exist $f^* \in {\R} \cup \{ -
\infty \}$ and $\tilde {\tilde \Omega} \in {\Cal F}$ such that

\itemitem{(a)$'$}  $P (\tilde {\tilde \Omega}) = 1, \theta (t, \cdot) (\tilde{ \tilde
\Omega}) \subseteq \tilde{ \tilde \Omega}$, $t \geq 0$;

\itemitem{(b)$'$}  $\tilde f(\omega) = f^* = \lim \limits _{t \to \infty}
(1/t) f(t, \omega)$ for every $\omega \in \tilde {\tilde \Omega}$.
 }

% \medskip   
 
\noindent
{\a Proof.}

{\d 
 
 [Mo], Stochastics, 1990, Lemma 7, pp\. 115--117.\qd
}

\bigskip

{\d

Proof of Theorem IV.2 is an application of Theorem IV.3. Requires Theorem IV.4 and the following sequence of lemmas.
}

\noindent
{\a Lemma 1}

{\sl  
 For each integer $k \geq 1$ and any $0 < a < \infty $,
%
$$ 
 E \sup _{0 \leq t \leq a} \| \phi (t,\o)^{-1} \|^{2k} < \infty;  
$$
%
\noindent
%
$$ 
E \sup _{0 \leq t_1, t_2 \leq a} \| \phi(t_2, \theta (t_1,\cdot)) \|^{2k} < \infty.  
$$
%
}
\noindent
{\a Proof.}

{\d 
Follows by standard sode estimates, the cocycle property for $\phi$ and 
H\"older's inequality. ([M], pp. 106-108).  \qd


\medskip

The next lemma is a crucial estimate needed to apply Ruelle-Oseledec theorem (Theorem IV.3).

}


\medskip
\noindent
{\a Lemma 2}
{\sl  
$$
     E \sup _{0 \leq t_1, t_2 \leq r} \log ^+ \| X(t_2,\cdot, \theta (t_1,
\cdot)) \| _{L (M_2)} < \infty.  
$$
%
\noindent

}

\newpage
\noindent
{\a Proof.}

{\d 

If $y(t,(v,\eta),\o)$ is the solution of the fde (8), then using Gronwall's
inequality, taking $E \displaystyle \sup_{0 \leq t_1,t_2 \leq r} \log^+ \displaystyle \sup_{\|(v,\eta)\| \leq 1}$ and applying Lemma 1, gives
%
$$
\align
E \sup_{0 \leq t_1,t_2 \leq r} \log^+ \sup_{\|(v,\eta)\| \leq 1}\, \| (y(t_2,(v,\eta), &\theta (t_1,\cdot)),y_{t_2}(\cdot,(v,\eta), \theta (t_1,\cdot)))\|_{M_2}\\
&< \infty.
\endalign
$$
%
Conclusion of lemma now follows by replacing $\o'$ with $\theta (t_1,\o)$ in the formula 
%
$$
  \align
   X(&t_2,(v, \eta),\o')\\
 &= (\phi (t_2,\o')(y(t_2,(v,\eta),\o')),\phi_{t_2} (\cdot, \o')\circ (id_J, y_{t_2} (\cdot, (v,\eta),\o')) 
\endalign
$$
%
\noindent
and Lemma 1.                         \qd

\medskip
\bigskip


The existence of the Lyapunov exponents is obtained by interpolating the discrete limit
%
$$
    \frac{1}{r}\, \lim _{k \to \infty} \frac {1}{k} \log \| X(kr, (v
(\omega), \eta(\omega)), \omega) \| _{M_2},   \tag{12}
$$
%
\noindent
$\hbox{ a\.a\. } \omega \in \Omega,\,\, (v, \eta) \in  L^2 (\Omega, M_2)$, between delay periods of length $r$. This requires the next two lemmas.
 

\bigskip
}

\newpage
\noindent
{\a Lemma 3}
{\sl  

 Let $h:\O \to \R^+$ be $\F$-measurable and suppose $ E \displaystyle\sup_{0\leq u \leq r} h(\theta (u,\cdot)$ is finite.
%
Then 
%
$$ \O_1 := \bigl ( \lim_{t \to \infty} \frac{1}{t} h (\theta (t,\cdot) =0 \bigr )$$
%
is a sure event and $\theta (t,\cdot) (\O_1) \subseteq \O_1$ for all $t \geq 0$.

}
%\bigskip
\noindent
{\a Proof.}

{\d  Use interpolation between delay periods and the discrete ergodic theorem applied to the $L^1$ function $$\hat h := \displaystyle\sup_{0\leq u \leq r} h (\theta (u,\cdot).$$
([Mo], Stochastics, 1990, Lemma 5, pp. 111-113.)  \qd

}

\bigskip
\noindent
{\a Lemma 4}

{\sl  

Suppose there is a sure event $\O_2$ such that 
$\theta (t,\cdot)(\O_2) \subseteq \O_2$ for all $t \geq 0$, and  
the limit (12) exists (or equal to $-\infty$) for all $\o \in \O_2$ and 
all $(v,\eta) \in M_2$. Then there is a sure event $\O_3$ such that  
$\theta (t,\cdot)(\O_3) \subseteq \O_3$ and 
%
$$
 \lim _{t \to \infty} \frac {1}{t} \log \| X(t, (v, \eta), \omega) \| _{M_2} =   \frac{1}{r}\, \lim _{k \to \infty} \frac {1}{k} \log \| X(kr, (v, \eta), \omega) \| _{M_2}, 
  \tag{13}
$$
%
\noindent
for all $\omega \in\Omega_3$ and all $\, (v, \eta) \in  M_2$.

}

\newpage
\noindent
{\a Proof:}

{\d Take $\O_3:= \hat \O \cap \O_1 \cap \O_2$. Use cocycle property for $X$, Lemma 2 and Lemma 3 to interpolate. ([Mo], Stochastics 1990, Lemma 6, pp. 113-114.)    \qd

}



\newpage
%\bigskip
\noindent
{\a Proof of Theorem IV.2.} (Sketch)

{\d

Apply Ruelle-Oseledec Theorem (Theorem IV.3) with

$T(\o):= X(r,\o) \in L(M_2)$, compact linear for $\o \in \hat \O$;

$\tau: \O \to \O; \quad \tau:=\theta (r,\cdot)$.

Then cocycle property for $X$ implies
%
$$ 
\align
X(kr,\o,\cdot)&= T(\tau^{k-1}(\o))\circ T(\tau^{k-2}(\o))\circ \cdots \circ T(\tau (\o))\circ T(\o)\\
&:= T^k (\o)
\endalign
$$
%
for all $\o \in \hat \O$.

Lemma 2 implies 
%
$$ E \log^+ \|T(\cdot)\|_{L(M_2)} < \infty.$$
%
Theorem IV.3 gives a random family of compact self-adjoint positive linear operators $\{ \Lambda (\o): \o \in \O_4 \}$ such that 
%
$$ \lim_{n \to \infty} [T^n (\o)^* \circ T^n (\o)]^{1/(2n)} := \Lambda (\o)$$
%
exists in the uniform operator norm and is a positive compact operator on $M_2$ for $\o \in \O_4$, a (continuous) shift-invariant set of full measure.
Furthermore each $\Lambda (\o)$ has a discrete spectrum
%
$$ e^{\mu_1 (\o)} >  e^{\mu_2 (\o)}  > e^{\mu_3 (\o)} > e^{\mu_4 (\o)} > \cdots$$
%
where the $\mu_i '$s are distinct, with no accumulation points except possibly $-\infty$. If  $\{\mu_i \}_{i=1}^{\infty}$ is infinite, then $\mu_i \downarrow -\infty$; otherwise they terminate at $\mu_{N(\o)}= -\infty$. If $\mu_i (\o) > -\infty$, then $e^{\mu_i (\o)}$ has finite multiplicity $m_i (\o)$ and finite-dimensional eigen-space $F_i (\o)$, with $m_i (\o):= dim F_i (\o)$.   Define 
%
$$  E_1 (\o):= M_2,\quad E_i (\o):= \bigl [\displaystyle \oplus_{j=1}^{i-1} F_j (\o) \bigr]^{\perp}, \quad
E_{\infty} (\o):= \hbox{ker}\,\, \Lambda (\o).$$
%
Then
%
$$ 
E_{\infty} (\o) \subset \cdots \subset E_{i+1}(\o)\subset E_{i} (\o) \cdots \subset E_2 (\o) \subset E_1 (\o) =M_2.
$$
% 
Note that $\hbox{codim}\, E_i (\o)= \sum_{j=1}^{i-1} m_j (\o) < \infty$. Also  
%
$$ 
\lim_{k \to \infty} \frac{1}{k} \log \|X (kr, (v,\eta), \o)\|_{M_2} =\cases  \mu_i (\o), \,\, \hbox{if} \,\, (v,\eta) \in E_i (\o) \backslash E_{i+1} (\o)\\
 -\infty \quad \hbox{if} \,\, (v,\eta) \in \hbox{ker}\,\, \Lambda (\o).
\endcases
$$
%
The functions 
%
$$ \o \mapsto \mu_i(\o),\quad \o \mapsto m_i (\o), \quad \o \mapsto N (\o)$$
%
are invariant under the ergodic shift $\theta (r,\cdot)$. Hence they take 
the fixed values $\mu_i, \,\, m_i,\,\, N$ almost surely, respectively. 

Lemma 4 gives a continuous-shift-invariant sure event $\O^* \subseteq \O_4$ such that
%
$$
\align 
\lim _{t \to \infty} \frac {1}{t} \log \| X(t, (v, \eta), \omega) \| _{M_2} &=   \frac{1}{r}\, \lim _{k \to\infty}\frac{1}{k} \log \| X(kr, (v, \eta), \omega) \|_{M_2}\\
&=\frac{\mu_i}{r} =: \lambda_i,  
\endalign
$$
%
\noindent
for  $\,(v, \eta) \in E_i (\o) \backslash E_{i+1} (\o),\,\, \omega \in \Omega^*, i \geq 1$.

$\{\lambda_i := \displaystyle\frac{\mu_i}{r}: i \geq 1 \}$ is the {\ii Lyapunov spectrum\/ } of (VIII). 

Since Lyapunov spectrum is discrete with no finite accumulation points, then  $\{ \lambda_i: \lambda_i > \lambda \}$ is finite for all $\lambda \in \R$.
 
To prove invariance of the Oseledec space $E_i(\o)$ under the cocycle $(X,\theta)$ use the random field
%
$$ 
\lambda ((v,\eta),\o):= \lim _{t \to \infty} \frac {1}{t} \log \| X(t, (v, \eta), \omega) \| _{M_2}  \qquad (v,\eta) \in M_2, \quad \o \in \O^*
$$
%
and the relations
%
$$ E_i (\o):= \{(v,\eta) \in M_2: \lambda ((v,\eta),\o) \leq \lambda_i \},$$
%
$$ \lambda (X(t,(v,\eta),\o),\theta (t,\o))= \lambda ((v,\eta),\o), \quad \o \in \O^*, \,\, t \geq 0$$
%
([Mo], Stochastics 1990, p. 122).   \qd

 }

\newpage
%\bigskip


{\d
     The non-random nature of the Lyapunov exponents $\{ \lambda _i
\}^\infty _{i = 1}$ of (VIII) is a consequence of the fact the $\theta$
is ergodic. (VIII) is said to be {\ii hyperbolic} if $\lambda _i
\ne 0$ for all $i \geq 1$.  When (VIII) is hyperbolic the flow satisfies
a {\ii stochastic saddle-point property} (or exponential dichotomy)
(cf\. the deterministic case with $E = C([-r,0], {\R}^d)$, $g_i \equiv 0$, 
$i = 1$, $\dots$, $m$, in Hale [H], Theorem
4.1, p\. 181).
 }

\bigskip
\noindent

 \noindent{\a  Theorem IV.5} {\ii (Random Saddles)}([Mo], Stochastics, 1990)

{\sl

Suppose the sfde (VIII) is hyperbolic.  Then there exist 

\item{(a)}  a set $\tilde \Omega ^* \in {\Cal F}$ such that  
$P(\tilde \Omega ^*) = 1$, and  
$\theta (t,\cdot) (\tilde \Omega ^*) = \tilde \Omega ^*$ for all $t \in {\R}$,


\item{} and

\item{(b)}  a measurable splitting
%
$$
     M_2 = {\Cal U} (\omega) \oplus {\Cal S} (\omega), \qquad \omega
\in \tilde \Omega ^*,
$$
%
\item{}  with the following properties:
\itemitem{(i)}  ${\Cal U} (\omega)$, ${\Cal S} (\omega)$, $\omega \in
\tilde \Omega ^*$, are closed linear subspaces of $M_2$, dim ${\Cal
U} (\omega)$ is finite and fixed independently of $\omega \in \tilde \Omega^*$.

\itemitem{(ii)}  The maps $\omega \mapsto {\Cal U} (\omega)$, $\omega
\mapsto {\Cal S} (\omega)$ are ${\Cal F}$-measurable into the
Grassmannian of $M_2$.

\itemitem{(iii)}  For each $\omega \in \tilde \Omega ^*$ and $(v,
\eta) \in {\Cal U} (\omega)$ there exists 
$\tau _1 = \tau _1 (v, \eta,\omega) > 0$ and a positive $\delta _1$, 
independent of $( v,
\eta,\omega)$ such that
%
$$
     \| X (t, (v, \eta), \omega) \| _{M_2} \geq \| (v, \eta) \|
_{M_2} e^{\delta _1 t}, \quad t \geq \tau _1.
$$
%
\itemitem{(iv)}  For each $\omega \in \tilde \Omega ^*$ and $(v,
\eta) \in {\Cal S} (\omega)$ there exists $\tau _2 = \tau _2 (v,\eta,\omega) >
 0$ and a positive $\delta _2$, independent of $(v,\eta,\omega)$ such that
%
$$
     \| X(t, (v, \eta), \omega) \| _{M_2} \leq \| (v, \eta) \| _{M_2}
e^{- \delta _2 t}, \quad t \geq \tau _2.
$$
%
\itemitem{(v)}  For each $t \geq 0$ and $\omega \in \tilde \Omega ^*$,
%
$$
     \align
     X(t, \omega, \cdot) ({\Cal U} (\omega)) &= {\Cal U} (\theta(t,
\omega)), \\
     X(t, \omega, \cdot) ({\Cal S} (\omega)) &\subseteq {\Cal S}
(\theta (t, \omega)).
     \endalign
$$
%
\itemitem{}  In particular, the restriction $$X(t, \omega, \cdot) \, |
\, {\Cal U} (\omega) : {\Cal U} (\omega) \to {\Cal U} (\theta (t,\omega ))$$ is a linear homeomorphism onto.
 
}

%\newpage


{\a Proof.}

[Mo], Stochastics, 1990, Corollary 2, pp. 127-130.  \qd
}

\newpage
\bigskip
\bigskip
\centerline{{\ii The Stable Manifold Theorem}}




\newpage

\noindent{\a 5.  Regular Linear Systems. Helix Noise }
{\d

\medskip

%
$$
     \left. \aligned
     dx(t) = \Bigl \{ \int_{[-r,0]} \nu (t) (ds) \,& x(t+s)\Bigr \} \,dt +
     dN(t) \, \int_{-r}^0 K(t)(s) \, x(t+s) \, ds\\
     &+ dL (t)\, x(t-), \quad  
     t > 0    \\
    (x(0), x_0)&=(v,\eta) \in M_2:= \R^d \times L^2 ([-r,0],\R^d)
     \endaligned
     \right \}
                                        \tag IX
$$
%
Linear systems driven by helix semimartingale noise, and memory driven by a 
measure-valued process $\nu$ on a complete filtered probability space
$(\O, \F, (\F_t)_{t \in \R}, P)$.
 }

\noindent{\a Hypotheses (C)}
{\d

\itemitem{(i)} The processes $\nu$, $K$ are stationary ergodic in
the sense that there is a measurable ergodic $P$-preserving flow
$\theta : {\R} \times \Omega \to \Omega$ such that for each $t
\in \R$, ${\Cal F}_t= \theta (t, \cdot)^{-1} (\Cal F_0)$ and
%
$$
     \align
     \nu (t, \omega) &= \nu (0, \theta (t, \omega)), \quad t \in {\R},\,\,
      \omega \in \Omega  \\
     K(t, \omega) &= K(0, \theta (t, \omega )), \quad t \in {\R},\,\,
\omega \in \Omega. 
     \endalign
$$
%
\itemitem{(ii)}  $L=M+V$, $M$ continuos local martingale, $V$ B.V. process. The processes $N$, $L$, $M$ have jointly stationary
ergodic increments:
%
$$
     \align
     N(t + h, \omega) - N(t, \omega)&= N(h, \theta (t, \omega)), \\
     L(t + h, \omega) - L(t, \omega )& = L(h, \theta (t, \omega)), \\
     M (t + h, \omega) - M(t, \omega)& = M(h, \theta (t, \omega)),
  \endalign
$$
%
\noindent
for $ t \in {\R},\,\, \omega \in \Omega$.  
 
\medskip
\bigskip

Semimartingales satisfying Hypothesis (C)(ii) were studied by  de
Sam Lazaro and P\.A\. Meyer ([S-M], 1971, 1976), \c Cinlar, Jacod, Protter and Sharpe [CJPS], Protter [P], 1986. 

     Equation (IX) is regular
w\.r\.t\. $M_2$ with a measurable flow $X: {\R}^+ \times M_2 \times \Omega
 \to M_2$.  This flow satisfies
Theorems III.4 and the cocycle property.  This is achieved via a construction in ([M-S], AIHP, 1996) based on the following consequence of Hypothesis (C)(ii):

\bigskip
\noindent
 }

\newpage
\noindent{\a  Theorem IV.6} ([Mo], Survey paper, 1992, [M-S], AIHP, 1996)  
 
{\sl
Suppose $M$ satisfies Hypothesis (C)(ii).  
Then there is  an $({\Cal
F} _t)_{t \geq 0}$-adapted version 
$\phi : {\R}^+ \times
\Omega \to {\R} ^{d \times d}$ of the solution to the matrix equation
%
$$
     \left. \aligned
     d \phi (t)& = dM(t)\phi (t) \quad t > 0 \\
     \phi (0)& = I \in {\R}^{d \times d}
     \endaligned
     \right \}
                                        \tag X
$$
%
\noindent
and a set $\Omega _1 \in {\Cal F}$ such that

\itemitem{(i)}  $P(\Omega _1) = 1$;

\itemitem{(ii)}  $\theta (t, \cdot) (\Omega _1) \subseteq \Omega _1$
for all $t \geq 0$;

\itemitem{(iii)}  $\phi (t_1 + t_2,\omega) = \phi (t_2, \theta
(t_1, \omega)) \phi (t_1, \omega)$ for all 
\itemitem{} $t_1, t_2 \in {\R}^+$ and every $\omega \in \Omega _1$;

\itemitem{(iv)}  $\phi (\cdot, \omega)$ is continuous for every
$\omega \in \Omega _1$.
 }
\bigskip    
 
{\d
A proof of Theorem IV.6 is given in ([Mo], Survey, 1992; [M-S], AIHP, 1996): either by a 
 double-approximation argument or via perfection techniques.

     The existence of a discrete non-random Lyapunov spectrum $\{
\lambda _i \}^\infty _{i = 1}$ for the sfde  (IX) is proved via Ruelle-Oseledec multiplicative ergodic
theorem which requires the integrability property (Lemma 2):
%
$$
     E \sup _{0 \leq t_1, t_2 \leq r} \log ^+ \| X(t_1, \theta (t_2,
\cdot), \cdot) \| _{L (M_2)} < \infty.  
$$
%
\noindent
The above integrability property is established under the
following set of hypotheses on $\nu$, $K$, $N$, $L$:


}
%\bigskip
%\newpage
\noindent
\noindent{\a Hypotheses (I)}

{\d
\item{(i)}  
%
$$
     \gathered
     \sup _{-r \leq s \leq 2r} \bigg | \frac {d \bar {\bar \nu}
(\cdot) (s)}{ds} \bigg | ^2, \quad \sup_{0 \leq t \leq 2r, -r \leq s
\leq 0} \| K(t, \cdot) (s) \| ^3, \\
     \sup _{0 \leq t \leq 2r, -r \leq s \leq 0} \| \frac
{\partial}{\partial t} K(t, \cdot) (s) \|^3, \quad \sup _{0 \leq t
\leq 2r, -r \leq s \leq 0} \| \frac {\partial}{\partial s} K(t,
\cdot) (s) \| ^3, \\
     \{ | V| (2r, \cdot) \}^4, 
     \endgathered
$$
%
\item{}  are all integrable, where 
$$\bar {\bar \nu}(\o)(A):= \int_0^{\infty} |\nu (t,\o)|\{(A-t)\cap [-r,0]\}\,dt,
\quad A \in Borel [-r,\infty)$$
has a locally (essentially) bounded density      $\displaystyle \frac {d \bar {\bar \nu}
(\cdot) (s)}{ds}$; and    
$| V|=$  total variation of
$V$ w\.r\.t\. the Euclidean norm $\| \cdot \|$ on ${\R}^{d \times
d}$.

\item{(ii)}  Let $N=N^0+ V^0$ where the local $({\Cal
F}_t)_{t \geq 0}$-martingale $N^0 = (N^0_{ij})^d_{i,j = 1}$ and the
bounded variation process 
\item{} $V^0 = (V^0_{ij})^d_{i, j = 1}$ are such
that 
%
$$
     \{ [ N^0_{ij}] (2r, \cdot) \}^2, \quad
     \{ | V^0_{ij} | (2r, \cdot) \}^4, \quad i, j = 1, 2, \dots, d
$$
%
\item{}  are integrable.   

$| V^0_{ij} | (2r, \cdot)=$  total
variation of $V^0_{ij}$ over $[0, 2r]$.

\item{(iii)}  $[ M_{ij}] (1) \in L^\infty (\Omega, {\R}), \quad i, j = 1, 2, \dots, d.$
%

  The  integrability property of the cocycle $(X,\theta)$  is a 
consequence of
%
$$
     E \log ^+ \sup _{0 \leq t_1, t_2 \leq r,\, \| (v, \eta) \| \leq 1}
| x (t_1,(v, \eta), \theta(t_2, \cdot) ) | < \infty.  
$$
%
\noindent
Proof of latter property uses lengthy argument based on
establishing the existence of suitable higher order moments for the
coefficients of an associated random integral equation.  (See Lemmas (5.1)-(5.5) in [M-S],I, AIHP, 1996.)

\medskip
    
 Since $\theta$ is ergodic, the multiplicative ergodic theorem
( Theorem IV.3, Ruelle) now gives a fixed discrete set of Lyapunov exponents
}
\bigskip
\noindent
\noindent{\a  Theorem IV.7} ([Mo], Survey, 1992; [M-S], AIHP, 1996)
 
{\sl
Under Hypotheses (C) \& (I), the statements of Theorems IV.2 and IV.5 hold
true for the linear sfde (IX).
 }
\bigskip
 
{\d   
     Note that the Lyapunov spectrum of (IX) does not change if one
uses the state space $ D([-r,0], {\R}^d)$ with the
supremum norm $\| \cdot \|_\infty$ ([M-S], AIHP 1996).
}

\end
\newpage