Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Stochastic classical solutions for space–time fractional evolution equations on a bounded domain

Space–time fractional evolution equations are a powerful tool to model diffusion displaying space–time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution equations featuring time-nonlocal initial conditions. We discuss the interpretation of the new stochastic representation. As part of the proof a new result about inhomogeneous Caputo evolution equations is proven.     

Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space

We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings.     

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.     

μ-average N-widths on the Wiener space

In this paper we investigate the asymptotic behaviour of μ-averagen-widths of integral operatorK on the Wiener space, whereK is the inverse operator of an ordinary linear differential operatorL of orderm. For 1≤p.q<∞ . and forp∈[1, ∞),q∈[2, ∞) . Supported by the Fund. of Dooctoral program of NECC.     

Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space

In this paper we define a more general convolution product of functionals on Wiener space and develop the fundamental relationships between the generalized Fourier–Feynman transform and the convolution product.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities

Let (W,H,μ) be an abstract Wiener space, assume that $\text{d}\text{ν=L}\text{d}\text{μ}$ is a second probability measures on $\text{(W,}\text{B}\text{(W))}$ such that $\text{L=}\text{1}\text{c}\text{exp}\text{?f}$, with $\text{f∈}\text{D}{}_{\mathrm{2,1}}$ lower bounded and H-convex. Let $\text{T=I}{}_{\mathrm{W}}\text{+??,}\text{?∈}\text{D}{}_{\mathrm{2,1}}$, be the solution of the Monge problem transporting μ to ν and realizing the H-Wasserstein distance between μ and ν. We prove that $\text{?∈}\text{D}{}_{\mathrm{2,2}}$ hence the Gaussian Jacobian $\text{Λ=}\text{det}{}_2\text{(I+?}{}^2\text{?)}\text{exp}\text{{}\text{L}\text{??1/2|??|}{}_{\mathrm{H}}{}^2\text{}}$ is well-defined and T is the strong solution of the Monge–Ampère equation ΛL°T=1 a.s. on W. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Subseries convergence in the space of functions of bounded φ-variation

Let there be given two F-spaces (X, ∥ ∥*), (Y, ∥ ∥*), X ? Y. A series Σ ₁ ^∞ x_i of elements in X is said to have property O(X, Y) (in the case when X = Y, ∥ ∥ = ∥ ∥*-property O) if perfect boundedness of this series in (X, ∥ ∥) implies its perfect (subseries) convergence in ( Y, ∥ ∥*). Conditions are determined for a series of elements in the space of functions of bounded φ-variation to have property O(X, Y). These conditions are rephrased for the case when convergence with respect to the norm ∥∥ _? ^v is replaced by modular convergence generated by φ-variation.     

Combinatorial proof of the inversion formula on the Kazhdan–Lusztig R-polynomials

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\) -polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.     

Besov–Lipschitz and Mean Besov–Lipschitz Spaces of Holomorphic Functions on the Unit Ball

We give several characterizations of holomorphic mean Besov–Lipschitz spaces on the unit ball in ${\mathbb C^N} $ and appropriate Besov–Lipschitz spaces and prove the equivalences between them. Equivalent norms on the mean Besov–Lipschitz spaces involve different types of L ^p -moduli of continuity, while in characterizations of Hardy–Sobolev spaces we use not only the radial derivative but also the gradient. The characterization in terms of the best approximation by polynomials is also given.     

Analog of the Cayley–Sylvester formula and the Poincaré series for an algebra of invariants of ternary form

An explicit formula is obtained for the number ν _d (n) of linearly independent homogeneous invariants of degree n of a ternary form of order d. A formula for the Poincaré series of the algebra of invariants of the ternary form is also deduced.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

The Shapley–Folkman theorem and the range of a bounded measure: an elementary and unified treatment

We present proofs, based on the Shapley–Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov’s celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley–Folkman theorem in mathematical economics.