Download PDF by Arieh Iserles: Acta Numerica 1998

By Arieh Iserles

ISBN-10: 0521643163

ISBN-13: 9780521643160

Acta Numerica is an annual quantity providing substantial survey articles in numerical research and medical computing. the topics and authors are selected by way of a unique foreign Editorial Board on the way to document an important and well timed advancements within the topic in a fashion available to the broader group of execs with an curiosity in medical computing.

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I −s (ei · ,β H 1,2 ) = e− 2s |β|H 1,2 . 2). From this we get easily the final result for I −s , by expanding the exponential in powers of λ and using dominated convergence to interchange summation and integrals. 12. 11, in the case where s ∈ R, s > 0, the functional I −s relative to H 1,2 ([0, t]; Rn ) is equal to: t 0 I −s e−λ V (γ(r)+x)dr f (γ(t) + x) = E e −λ t 0 V b(r) √ +x s dr f b(t) √ +x s , where E is the expectation with respect to Brownian motion b(r), r ∈ [0, t]. This is an easy consequence of the relation between I −s and E we proved before, when we showed that N (0; Q), Q(s, t) = s ∧ t, is a realization of Wiener measure.

K 2 2 Hence μ({y ∈ RN | k |yk | dμ(y) < ∞}) = 1 and μ is supported by l . Denoted −1 by ν = γ ◦ μ, it is easy to see that ν = N (a, Q) on H. Let us try to consider the same object for Q replaced by 1lH . For simplicity of notation let us take a = 0. Let us consider the product measure μ := ⊗∞ k=1 N (0, 1k ) on RN . l2 as a subset of RN belongs to the σ-algebra σ(Z) generated by the cylinder subsets of RN , since n N y∈R | 2 l = p∈N N ∈N n>m≥N yk2 < m 1 p Oscillatory and Probabilistic Integrals 31 as seen by realizing that y ∈ l2 iff for any > 0 there exists an N ∈ N such that n for any n > m ≥ N , one has m yk2 < , and taking = 1/p, p ∈ N.

L2 as a subset of RN belongs to the σ-algebra σ(Z) generated by the cylinder subsets of RN , since n N y∈R | 2 l = p∈N N ∈N n>m≥N yk2 < m 1 p Oscillatory and Probabilistic Integrals 31 as seen by realizing that y ∈ l2 iff for any > 0 there exists an N ∈ N such that n for any n > m ≥ N , one has m yk2 < , and taking = 1/p, p ∈ N. We shall now prove that l2 has μ-measure equal to 0. Let us consider the real-valued random variables Xk (y) := yk , k ∈ N on (RN , σ(Z), μ). We have that Xk ∈ L2 (RN , σ(Z), μ) and RN Xk (y)2 μ(dy) = 1.

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Acta Numerica 1998 by Arieh Iserles


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