By Alexander J. W.

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As it will explained, this problem is completely solved using the differentiation in the sense of distributions. Afterwards we give a straightforward application of this result to prove a 0 - 1 law for the Wiener measure. At the second section we construct the composition of the tempered distributions with nondegenerate Wiener functionals as Meyer-Watanabe distributions. This construction carries also the information that the probability density of a nondegenerate random variable is not only infinitely differentiable but also it is rapidly decreasing.

Hence we have proven: Lifting of S'(~ d) 48 L e m m a 1: The following representation holds in S(Ra): f = JR" f(x)C~dx. ) denotes the bilinear form of duality between D' and D. Proof: Let p~ be a mollifier. :(y)d~ = = /p,(y)f(y+F)dy ,_~ f(F). a > f(y)dy = < f(F),~ > = E[f(F):]. Corollary: QED We have (E~(F), 1) = d(dx#) ) 1 7 . , the probability density of F is not only C ~ but it is also a rapidly decreasing function). ~] = s (PF,~,,T)S'. ~(F), 1)f(y)dy. J QED 49 Remark: From the disintegration of measures, we have / E[~tF = x]f(x)dx = E[f(F)4a] = / f(x)(Ex(F),ia)dx hence E[@F = x] = (,~:(F), @ dx-almost surely.

Hence 7~j E D. T h e n from the d o m i n a t e d convergence theorem we have t 7 i j ---* "Yij in L p as well as V k 7ij~ LP ~.. Vl~/ij (this follows again "from 7" "at = ld). QED Lemma2 Let G E D . l,(G)] where G ~ li(G) is linear and for any 1 < r < g < oo, sup Illi(G)llr < + o o . G] = E [ f ( F ) . ik (G)] and sup lit,, ,k(a)llr < ~ . f o F ) = ES, f ( F ) V F ~ ~ ( V ( / o r ) , VFj) = Ea,jO, f ( F ) . Since o" is invertible, we obtain: O,1'(F) = ~ ~,i(v(/o F), VFj). G] J i hence we see that/i(G) I,(G) = = E j 5{VFjTqG}.

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A Lemma on Systems of Knotted Curves by Alexander J. W.


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