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By D. R Bland

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L. Rozovskii and the last limit exists and is finite. 23) it suffices to show that J = 0. It is convenient to assume that the basis {hi } is formed from elements of the set ΛV . Since the latter is dense in H, this last assumption involves no loss of generality. Moreover, it is obvious that for every r ≥ 1 there exists a function v˜r (t) continuous in the norm of V such that Λ˜ vr (t) = πr h(t) for all (t, ω). 18 the (i) functions v˜r are constructed on the basis of v. 21 for tnj+1 ≤ t and every ϕ ∈ V there is the equality tn j+1 ˜ n ) − h(t ˜ n ) − h(tn ) − h(tn )) Λϕ = (h(t j+1 j j+1 j ϕv ∗ (u)du, tn j we find easily for any r ≥ 0 t J = lim n→∞ − lim n→∞ (vn(2) (u) − (vn(1) (u))v ∗ (u)du − lim n→∞ 0 tn j+1 ≤t t 0 (2) (1) (˜ vr,n (u) − (˜ vr,n (u))v ∗ (u)du ˜ n ) − h(t ˜ n )) − (h(tn ) − h(tn )) (1 − πr )(h(tn ) − h(tn )).

4) also belongs to the basic results of this section. 5. 11. 4), and u = u(t) is its continuous modification in H. Then u(t) is a Markov random variable. 12. With no loss of generality, we can take K = 0 in conditions (A2 ) and (A3 ). 4) with A replaced by e−Kt (A − KI), where I is the identity operator, and with B replaced by e−Kt B, and these new A and B satisfy conditions (A2 ) and (A3 ) with K = 0. 13. Our assumption that the spaces in question are real is not essential and may be relaxed if, in conditions (A2 )–(A3 ), in place of vA(v1 + λv2 ), (v1 − v2 )(A(v1 ) − A(v2 )), vA(v) we write Re vA(v1 + λv2 ) , Re (v1 − v2 )(A(v1 ) − A(v2 )) , Re vA(v)).

L. Rozovskii for all i, j and any basis {hi }, where dP × d x ∞ µ(A) = E is the differential of the measure t χA (t, ω)d x t , 0 defined on the product of F and the Borel σ-algebra on [0, ∞). We call the process Qx the correlation operator of x. ) for any t ≥ 0, then there exists a square-integrable martingale y(t) in E which is strongly continuous in t and such that, for every orthonormal basis {hi } and every v ∈ E, T ≥ 0, n lim E sup v y(t) − n→∞ t≤T 2 t vB(s)hi d(hi x(s)) = 0. s. for all t.

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