The MET for Related Linear and Affine RDS
Let V be a finite-dimensional vector space, and A: V → V be a linear operator. Then the standard constructions of linear algebra yield A −1 on V, A* on the dual space V*, ∧ k A on ∧ k V (the k-th exterior power of V where 1 ≤ k ≤ d), in particular ∧d A =
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Summary Let V be a finite-dimensional vector space, and A : V -+ V be a linear operator. Then the standard constructions of linear algebra yield A - l on V, A* on the dual space V*, 1\k A on 1\kV (the k-th exterior power of V where 1 ~ k ~ d), in particular 1\d A= det A on 1\dV. If U c V is A-invariant, A induces operators Alu on U, A~ on the quotient space u~ = VjU and Aj_ on the orthogonal complement U j_. Further, if B : W -+ W is another linear operator, we have A EBB on the direct sum V EB W and A Q9 B on the tensor product V Q9 W. Assume now that - 1 , use d> = A and transposition to arrive at d>*- 1 = -A**- 1. Formally the same calculation goes through for a Stratonovich SDE. The cocycle cP(t, w) := on U (·), if>u(t,w) := if>(t,w)lu(w): U(w)-+ U(8(t)w)
is a linear co cycle on the bundle U ( ·). We define a second linear cocycle if>~ on the quotient space U(w)~ ·ffi.d /U (w) of equivalence classes (with respect to the equivalence relation x '"""' y {::=:::} x- y E U(w)) by U(w)~ ::1
[x]w t-+
if>~(t, w)[x]w :=
[if>(t, w)x]e(t)w E
U(8(t)w)~,
where [x]w is the equivalence class containing x E ffi.d over the fiber w. Finally, let U(w).L be the orthogonal complement of U(w) in ffi.d (with respect to the standard scalar product) 3 and let the cocycle if>.L on the linear bundle U(·).L be defined by U(w).L ::1 x t-+ if>.L(t,w)x := (if>(t,w)x)*(t)w E U(8(t)w).L,
where y t-+ y-(; = 1r.L(w)(y) is the orthogonal projection onto U(w).L over w. The restriction of the canonical projection 1r(w): ffi.d-+ U(w)~, x t-+ [x]w, to U(w).L defines an isomorphism of U(w).L and U(w)~, which is an isometry if we define a scalar product on U(w)~ by ([x]w, [Y]w) := (xt, y"(;). With this done we can identify the co cycles if>.L on U (-) .L and if>~ on U ( ·) ~ for our purposes. 3
Note that U(w).l. is in general not invariant with respect to .i, d1)z=l.. .. ,p} and splitting E 1 , ..• ,Ep· Assume that the random lineaT s-ubspace U(w) c ffi.d is non-trivial and iP-invariant. Then: (i) The cocycle iPu = iP Iu (.) satisfies the integrability conditions of the MET and has iipfctrum
and splitting
Ek,
n U, ... , Ekr n U,
wheTe 1 ::; k: 1 < k: 2 < ... < k:r ::; p an; the indices joT which df U) > 0. In particular, U(w) has the form
:=
dim(Ek
n
U(w) = (ii) The cocycle iP~ on u~ (equivalently, the cocycle iPj_ on U j_) satisfies the IC of the MET and has spect·rum
and splitting where
E;n,
n U j_, ... , E;, n U j_, 4
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Chapter 5. MET for Related Linear RDS
E!,i = (EBk~i(Ff n Uj_))j_ and 1 :S m 1 < ... < mq :S p are the indices for which di: := dim( Ff n U j_) n Uj_) > 0. (iii) We have s(e, (t, w) on 1\kJRd, 1 :::;: k :::;: d, satisfies the IC of the respective MET and the MET holds on the same forward invariant (resp. invariant) set f2. In particular, lim ((/\ktJ>(t,w))*(/\ktJ>(t,w))) 112t = 1\k(w)
t--+co
{(>.~k),d~k))i=l, ... ,p(k)} is determined as follows: The )..~k) are the different numbers in the list of (~) numbers and the spectrum S((J,f\ktJ>)
=
where this list star-ts with >.ik) = A1 +· · ·+Ak and ends with)..~
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