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On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

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Abstract

Let w and µ be respectively the conditional Wiener measure in C 0([0, 1]) and the centered Gaussian measure in L 2[0, 1] with the correlation operator (−d 2/dx 2)−1. We prove the equivalence of these two measures in the following sense: for any Borel set A ⊂ L 2[0, 1] the set A ∩ C 0([0, 1]) is a Borel subset of C 0([0, 1]) and µ(A) = w(A∩C 0([0, 1])).

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  1. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna (Moscow region), Russia

    Peter Zhidkov

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  1. Peter Zhidkov
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Correspondence to Peter Zhidkov.

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Zhidkov, P. On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure. Rend. Circ. Mat. Palermo 58, 427–440 (2009). https://doi.org/10.1007/s12215-009-0033-z

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  • DOI: https://doi.org/10.1007/s12215-009-0033-z

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