Local invertibility of adapted shifts on Wiener space,under finite energy condition

In this paper we study the connection between local invertibility and global invertibility of adapted shifts on Wiener space. First we go from the global to the local and we obtain an explicit formula for the inverse of a stopped adapted shift which is invertible. Then we take the opposite direction and we show that under finite energy condition, a shift which is locally invertible is also invertible. We work in a general framework which applies both to the classical Wiener space and to the Brownian sheet.     

Flows associated to Cameron-Martin type vector fields on path spaces over a Riemannian manifold

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.     

Quasi-Invariance of the Wiener Measure on Path Spaces: Noncompact Case

For a geometrically and stochastically complete, noncompact Riemannian manifold, we show that the flows on the path space generated by the Cameron-Martin vector fields exist as a set of random variables. Furthermore, if the Ricci curvature grows at most linearly, then the Wiener measure (the law of Brownian motion on the manifold) is quasi-invariant under these flows.     

Semi-complete foliations associated to Hamiltonian vector fields in dimension 2

We classify the foliations associated to Hamiltonian vector fields on C², with an isolated singularity, admitting a semi-complete representative. In particular we also classify semi-complete foliations associated to the differential equation .     

Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).     

The average errors for lagrange interpolation on the Wiener space

For the weighted approximation in L _p -norm, we determine the asymptotic order for the paverage errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.     

Complete polynomial vector fields on the complex plane

We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.     

Invertibility of the Gabor frame operator on the Wiener amalgam space

We use a generalization of Wiener's 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W(L^∞,?¹)(R^d), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the canonical dual belongs also to W(L^∞,?¹)(R^d).     

Invariant monotone vector fields on Riemannian manifolds

Various concepts of invariant monotone vector fields on Riemannian manifolds are introduced. Some examples of invariant monotone vector fields are given. Several notions of invexities for functions on Riemannian manifolds are defined and their relations with invariant monotone vector fields are studied.     

Conformal vector fields on spacetimes

We study conformal vector fields and their zeros on spacetimes which are non-conformally-flat. Depending on the Petrov type, we classify all conformal vector fields with zeros. The problems of reducing a conformal vector field to a homothetic vector field are considered. We show that a spacetime admitting a proper homothetic vector field is (locally) a plane wave. This precises a well-known theorem of {Alekseevski}, where all these spacetimes are determined in a more general form.     

The Yang-Mills fields on the Minkowski space

Let the coordinatex=(x ⁰,x ¹,x ²,x ³) of the Minkowski spaceM ⁴ be arranged into a matrix

Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.     

On the domain invariance of countably condensing vector fields

Using homotopy theory, we give the domain invariance theorem for countably condensing vector fields, where the notion of countably condensing maps is due to Väth. A starting point of this investigation is that there is a symmetric characteristic set for a countably condensing map.     

Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis

Extensions of the Nourdin-Peccati analysis to Rn-valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein's lemma), to classical analysis on Wiener space. Partial differential equations for the density of an Rn-valued centered random variable Z=(Z¹,…,Zn) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix (−DL−1Zi|DZj), i,j=1,2,…,n, where D and L are respectively the derivative operator and the generator of the Ornstein-Uhlenbeck semigroup on Wiener space.     

Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms

In this paper, we study real hypersurfaces all of whose integral curves of characteristic vector fields are plane curves in a nonflat complex space form.      

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.     

The Dirac spectrum on manifolds with gradient conformal vector fields

We show that the Dirac operator on a spin manifold does not admit L² eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Asymptotic stability at infinity for differentiable vector fields of the plane

Let be a differentiable (but not necessarily C¹) vector field, where σ>0 and . Denote by R(z) the real part of z∈C. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector v∈R² the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R²∪{∞} is a repellor (respectively an attractor) of the vector field X+v.