Solution of the Monge-Ampère equation on Wiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge-Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400-428], w.r.t the Wiener measure is also the strong solution of the Monge-Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.     

Gas-phase infrared photodissociation spectroscopy of tetravanadiumoxo and oxo-methoxo cluster anions.

The infrared spectra of the binary vanadium oxide cluster anions V(4)O(9)(-) and V(4)O(10)(-) and of the related methoxo clusters V(4)O(9)(OCH(3))(-) and V(4)O(8)(OCH(3))(2)(-) are recorded in the gas phase by photodissociation of the mass-selected ions using an infrared laser. For the oxide clusters V(4)O(9)(-) and V(4)O(10)(-), the bands of the terminal vanadyl oxygen atoms, nu(V-O(t)), and of the bridging oxygen atoms, nu(V-O(b)-V), are identified clearly. The clusters in which one or two of the oxo groups are replaced by methoxo ligands show additional absorptions which are assigned to the C-O stretch, nu(C-O). Density functional calculations are used as a complement for the experimental studies and the interpretation of the infrared spectra. The results depend in an unusual way on the functional employed (BLYP versus B3LYP), which is due to the presence of both V-O(CH(3)) single and V=O double bonds as terminal bonds and to the strong multireference character of the latter.     

Gas phase oxidation of alkoxo ligands in bis(peroxo)[MO(O2)2(OR)]- and trisoxo [MO3(OR)]- anions (M = Cr, Mo, W)

The anions [M(VI)O(O(2))(2)(OR)](-) and [M(VI)O(3)(OR)](-)(M = Cr, Mo, W; R = H, Me, Et, (n)Pr, (i)Pr) were transferred to the gas phase by the electrospray process. Their decomposition was examined by multistage mass spectrometry and collisional activation experiments. The molybdate and tungstate anions [M(VI)O(O(2))(2)(OR)](-) underwent parallel elimination of aldehyde (ketone) and dioxygen while the equivalent chromate underwent loss of dioxygen only. The peroxo ligands were the source of oxidising equivalents in both reactions. For each alkoxo ligand, the total yield of aldehyde for the tungstate system exceeded that for the molybdate system. Collisional activation of [M(VI)O(3)(OMe)](-) led to clean elimination of formaldehyde with the metal centre supplying the oxidising equivalents. For larger alkoxo ligands, only the chromate centre eliminated aldehyde, while the molybdate and tungstate centres underwent clean loss of alkene. Threshold activation voltages indicated that the peroxo ligands of [W(VI)O(O(2))(2)(OMe)](-) are more oxidising than the tungstate centre of [W(VI)O(3)(OMe)](-). (2)H and (18)O isotope tracing experiments were consistent with a formal hydride transfer mechanism operating for oxidation of alkoxo ligand in each system. In the solid state, anions [M(VI)O(O(2))(2)(OR)](-) are typically pentagonal pyramidal (oxo in apical site) while [M(VI)O(3)(OR)](-) are tetrahedral. The data indicate that an equatorial ligand position is the site of alkoxo oxidation in [M(VI)O(O(2))(2)(OR)](-) anions. Comparisons of the gas phase data with those for a solution phase system are made.     

Hausdorff measures on the Wiener space

We construct a Hausdorff measure of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Krée-Watanabe theorem concerning the regularity of the images of the Wiener measure.     

Measure transport on Wiener space and the Girsanov theoremTransport de mesure sur l'espace de Wiener et théorème de Girsanov

Let (W,H,μ) be an abstract Wiener space, and assume that ν_i, i=1,2, are two probability measures on $\text{(W,}\text{B}\text{(W))}$ which are absolutely continuous with respect to μ. Assume that the Wasserstein distance between ν₁ and ν₂ is finite. Then there exists a map T=I_W+ξ of W into itself such that ξ:W→H is measurable and 1-cyclically monotone such that the image of ν₁ under T is ν₂. Moreover T is invertible on the support of ν₂. We give also some applications of this result such as the existence of the solutions of the Monge–Ampère equation in infinite dimensions. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1025–1028.