Path-dependent SDEs in Hilbert spaces

We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, G\^ateaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the G\^ateaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions

Supergravity with Self-dual B fields and Instantons in Noncommutative Gauge Theory

We study Type IIB supergravity in the presence of (euclidean) D3 branes and nonzero self-dual B-fields. We point out that the Einstein frame metric is identical to the full geometry for D3 branes without B fields turned on. Furthermore, in a decoupling limit in which the theory is conjectured to be dual to noncommutative Yang-Mills theory, the entire Einstein metric remains intact, and in particular, is asymptotically flat. We construct D-instanton solutions in this geometry. We show that in the decoupling limit the D-instanton action agrees with the action of the corresponding instanton in the noncommutative Yang-Mills theory and is expressed in terms of the open string coupling. Some other aspects of this correspondence, which have unusual features because the underlying metric is asymptotically flat, are explored.Comment: 25 pages, harvma

Entropy of Anisotropic Universe and Fractional Branes

We obtain the entropy of a homogeneous anisotropic universe applicable, by assumption, to the fractional branes in the universe in the model of Chowdhury and Mathur. The entropy for the 3 or 4 charge fractional branes thus obtained is not of the expected form E^{{3/2}} or E^2. One way the expected form is realised is if p \to \rho for the transverse directions and if the compact directions remain constant in size. These conditions are likely to be enforced by brane decay and annihilation, and by the S, T, U dualities. T duality is also likely to exclude high entropic cases, found in the examples, which arise due to the compact space contracting to zero size. Then the 4 charge fractional branes may indeed provide a detailed realisation of the maximum entropic principle we proposed recently to determine the number (3 + 1) of large spacetime dimensions.Comment: Version 2: 21 pages. More discussion and references added. To appear in General Relativity and Gravitatio