By Shiing-Shen Chern, Stephen Smale (ed.)

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Example text

Moreover, one can compute the moments of the exponential of this integral in the usual way and conclude that the family is uniformly integrable. 6). A similar argument shows that the finite dimensional distributions also converge. 32). As above, we only argue the case of the single distribution of u(t, x) for t and x fixed. T h e desired conclusion follows from the fact that: Mt = u(i, x) - u0(x) - / [A + r i ] u ( s , • )(x) ds Jo is a martingale and t h a t : [ M , M ] t = 1^(0) f'lufaxtfds. e.

The limiting procedure is controlled by an estimate of the Poisson kernel. e. 7Trx{y)dt) = lPx{XTr = y , Tr£dt] and more precisely in its asymptotic behavior as r —» oo. Using the independence of the times of j u m ps and the sites of the lattice visited by the random walk and using standard properties of Poisson processes one gets: L e m m a I I . 1 . <) = Ej^Yy^^r^ri^y) k>r ^ ' where Nr(x)y) denotes the numbers of paths 7 of nearest neighbors = x, | 7 ( 1 ) | < r , • • • , |7(* — 1)| < r and 7 (fc) = y.

43) RENE A. A. 36). 45) with the initial condition m p (0, • ) = 1. It is convenient to write A x = AX1 + • • • AXp and: Vp(x) = J2 l 2 and Vi(x) = 0. 45) becomes: dmP „ , . -gf = HP(K)rnp where the operator HP(K) — /cAx + V(x) is a p-particle Schrodinger operator on the lattice 2Lpd. Proof: The proof we give does not use the explicit form of the moments as given by the Feynman-Kac formula. See the remark below. The proof is slightly longer than the proof discussed in the remark below but it is extremely useful in situations where the Feynman-Kac formula does not hold in a nice form.

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