Transformation of Measure on Wiener Space

This paper considers the model of an arbitrarily distributed signal x observed through an added independent white Gaussian noise w, y = x + w. New relations between the minimal mean square error of the non-causal estimator and the likelihood ratio between y and w are derived. This is followed by an extended version of a recently derived relation between the mutual information I(x; y) and the minimal mean square error. These results are applied to derive infinite dimensional versions of the Fisher information and the de Bruijn identity. A comparison between the causal and non-causal estimation errors yield a restricted form of the logarithmic Sobolev inequality. The derivation of the results is based on the Malliavin calculus.

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“…The following close (but weaker) statement was obtained in the recent paper [10]: if f is a function measurable with respect to a Gaussian measure p on a separable Banaeh space X defined on a #-measurable set A and satisfying there the Lipschitz condition along the Cameron-Martin space E, then there exists a p-measurable function on X satisfying the same condition and coinciding with f #-almost everywhere. This statement follows from the theorem above, since f has a Borel modification satisfying the same condition on a Boret subset B C A with p(B) = #(A).…”

Section: If(= + H) -/(=)1 -< Llhllmentioning

confidence: 78%

On the extension of generalized Lipschitz mappings

Kobanenko

1998

Math Notes

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Let X be a locally convex topological space endowed with the Gaussian measure #; denote by E the generating kernel of the measure #, by Y a Banach space with norm [ 9 [, and let f: X --~ Y be a measurable mapping.According to the result from [1] (which is, in particular, a generalization of the well-known Fernique Theorem [2]) the foUowing property of f,implies the inclusion exp(el/I) ~ Lt(~,) for c > 0 small enough.Several problems related to this result arise. One of these problems is the extendability of a measurable mapping defined on a measurable subset in X and satisfying the requirement (1) on this set to the whole space X, preserving measurability and property (1) with the same constant. Note that a lot of papers are devoted to the study of the more special problem about the extendability of Lipschitz (or H61der) mappings f defined on a subset A of a normed (or even metric) space X and taking values in a normed space Y. One of the first results in this direction was obtained by McShane in [3]. He proved that any number-valued Lipschitz function on an arbitrary subset of a metric space X can be extended to the whole space while preserving the Lipschitz constant (the corresponding extension is given by a simple explicit formula). In the multidimensional case, the situation is more complicated. For example (see [4][5][6]), it can happen that both X and Y are Banach spaces, but there is a mapping f: A --* Y without a Lipschitz extension (even without any restriction on the value of the Lipschitz constant). For some pairs X and Y, an extension not increasing the Lipschitz constant always exists; and for some pairs of spaces, the extension, although it always exists, can have a greater Lipschitz constant (see [4,5]). One of the better known positive results is the theorem from [7], which states that each Lipschitz mapping [ defined on a subset of a Hilbert space X and taking values in a Hilbert space Y has a Lipschitz extension to the whole space X with the same Lipschitz constant. However, in contrast to scalar-valued functions, such extensions have no explicit description; their existence is deduced from a property of intersection of convex bodies. The result below is a generalization of the McShane theorem for the situation described above. It seems that the corresponding analog of the theorem from [7] for infinite-dimensional mappings is also true; however I did not succeed in proving this even for mappings to R 2 (the infinite-dimensional case can be handled if we prove E-Lipschitz extendability with the same Lipschitz constant for mappings to R").Recall that the images of complete separable metric spaces under continuous mappings to Hausdorff spaces are called SualiTt spaces (see [8]). The continuous images of Borel sets in complete separable metric spaces are also known to be Suslin. Most of the locally convex spaces that one meets in applications are also Suslin (although they can be nonmetrizable as, for example, the spaces S'(R"), :D(R"), D'(R") and separable Banach spaces with the weak topology).Deno...

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Transformation of Measure on Wiener Space

Cited by 120 publications

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Covariance identities and mixing of random transformations on the Wiener space

Covariance identities and mixing of random transformations on the Wiener space

On Mutual Information, Likelihood Ratios, and Estimation Error for the Additive Gaussian Channel

On the extension of generalized Lipschitz mappings

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