On Measures Equivalent to Wiener Measure

Wiener measure and Donsker’s theorem

R-bounded Fuzzy Measures Are Equivalent to Ε-possibility Measures

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe nonprobabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which μ(A ∪ B) = max(μ(A), μ(B)). In this paper, we show that possibility measures are, in some sense, universal approximat...

Measure Theory on Hausdorff Measures

Let y = h(x) be defined for 0 < x < oo and assume values in 0 ^ y ^ + co. Let S be any linear set of points and p an arbitrary positive number. Cover S by a countable number of open intervals h , I2, • • • of lengths xx, x2 , • • • each of which is less than p, and denote by mp(S; h) the lower bound of h(xi) + h(x2) + • • • for all such coverings of S. Then m(S; h) = limp|0mp(/S; h) is called t...

Wiener Way to Dimensionality

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...

Measure invariance on the Lie-Wiener path space

In this paper we extend some recent results on moment identities, Hermite polynomials, and measure invariance properties on the Wiener space, to the setting of path spaces over Lie groups. In particular we prove the measure invariance of transformations having a quasi-nilpotent covariant derivative via a Girsanov identity and an explicit formula for the expectation of Hermite polynomials in the...