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Equation. 1S + vVS - -Ivl 2 2 2 - V(t ,x)< o. Using the operator D*, this inequality is equivalent to D*S(t,Xn::; ~lvl2 + V(t,xn. Using the Dynkin formula for the function S(x, t) we have S(h,xd = EtlS(to,XJo ) +Etl ::; -Etl wo(XJo ) + Eh i tl D*S(s,X~)ds to itl to ~lvl2 + V(s,X~) ds, therefore Let us consider u(t,x) = VS(t,x). VS(t,x) - 1 21vl2 - V(t,x) reaches its maximum value at u( t, x). 2 which minimizes the functional S, and u( t) its drift. e. their drift may depend on the future). Each process X t E F defines a law on the path space and by Girsanov theorem the laws on OX!

W(q) = sup{w(p);p::; q,p compact projection} for all open projections q). This is proved for unital C*-algebras in [1]. Suppose 1 rf- U, and let p be a projection in U**. It is known that p is open if and only if p is open in U** (~ U** EB C), and that p is compact if and only if p is closed in U** ([6]). Then, working in U** with the canonical extension w makes the job. e. w(p) = inf{w(q); q ~ p, q open projection} for all closed projections p). We can now state the following extensions of classical "portemanteau" theorems.

M. Chebotarev, S. Yu. 2000, N 1645-00. [18] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin (1976). [19] R. , N-Y. (1978). [20] M. Reed, B. , (1981). [21] D. Kilin and M. Schreiber, "Influence of phase-sensitive interaction on the decoherence process in molecular systems", LANL preprint, quant-ph/9707054 (1997). [22] S. Schneider, G. J. Milburn, "Decoherence in ion traps due to laser intensity and phase fluctuations", LANL preprint quant-ph/9710044 (1997). [23] L. Lanz, O.

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