scholarly journals SMOOTH PROBABILITY MEASURES AND ASSOCIATED DIFFERENTIAL OPERATORS

1999 ◽  
Vol 02 (01) ◽  
pp. 51-78 ◽  
Author(s):  
O. G. SMOLYANOV ◽  
H. v. WEIZSÄCKER
Keyword(s):  
Differential Operators ◽  
Finite Dimensional ◽  
Differentiable Measure ◽  
Chaos Decomposition ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Associated Differential Operators ◽  
Locally Convex ◽  
Nonsmooth Mappings

We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the L2-space of a differentiable measure the analog of the classical concepts of gradient, divergence and Laplacian (which coincides with the Ornstein–Uhlenbeck operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite straight forward and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the procedure of quantization of anharmonic oscillators is discussed.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

scholarly journals Generalized functionals of Brownian motion

1994 ◽  
Vol 7 (3) ◽  
pp. 247-267
Author(s):  
N. U. Ahmed
Keyword(s):  
Brownian Motion ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Broad Class ◽  
Wiener Measure ◽  
Multiple Stochastic Integrals ◽  
Finite Dimensional ◽  
Nonlinear Functionals ◽  
Recent Developments ◽  
Locally Convex

In this paper we discuss some recent developments in the theory of generalized functionals of Brownian motion. First we give a brief summary of the Wiener-Ito multiple Integrals. We discuss some of their basic properties, and related functional analysis on Wiener measure space. then we discuss the generalized functionals constructed by Hida. The generalized functionals of Hida are based on L2-Sobolev spaces, thereby, admitting only Hs, s∈R valued kernels in the multiple stochastic integrals. These functionals are much more general than the classical Wiener-Ito class. The more recent development, due to the author, introduces a much more broad class of generalized functionals which are based on Lp-Sobolev spaces admitting kernels from the spaces 𝒲p,s, s∈R. This allows analysis of a very broad class of nonlinear functionals of Brownian motion, which can not be handled by either the Wiener-Ito class or the Hida class. For s≤0, they represent generalized functionals on the Wiener measure space like Schwarz distributions on finite dimensional spaces. In this paper we also introduce some further generalizations, and construct a locally convex topological vector space of generalized functionals. We also present some discussion on the applications of these results.

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

WEAK CONVERGENCE OF NONLINEAR TRANSFORMATIONS OF INTEGRATED PROCESSES: THE MULTIVARIATE CASE

2009 ◽  
Vol 25 (5) ◽  
pp. 1180-1207 ◽  
Author(s):  
Norbert Christopeit
Keyword(s):  
Weak Convergence ◽  
Recent Work ◽  
Invariance Principle ◽  
Nonlinear Transformations ◽  
New Class ◽  
Triangular Arrays ◽  
Integrable Functions ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Integrated Processes

We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p-integrable functions) is introduced that adapts the concept of locally integrable functions in Pötscher (2004, Econometric Theory 20, 1–22) to the multidimensional setting.

scholarly journals A note on(gDF)-spaces

2000 ◽  
Vol 23 (5) ◽  
pp. 367-368
Author(s):  
Renata R. del-Vecchio ◽  
Dinamérico P. Pombo ◽  
Cybele T. M. Vinagre
Keyword(s):  
Locally Convex Spaces ◽  
Finite Dimensional ◽  
Locally Convex ◽  
Convex Spaces

Certain locally convex spaces of scalar-valued mappings are shown to be finite-dimensional.

Diffusion MRI in Peripheral Nerves: Optimized b Values and the Role of Non-Gaussian Diffusion

Radiology ◽  
2021 ◽  
Author(s):  
Olivia Foesleitner ◽  
Alba Sulaj ◽  
Volker Sturm ◽  
Moritz Kronlage ◽  
Tim Godel ◽  
...  
Keyword(s):  
Peripheral Nerves ◽  
Diffusion Mri ◽  
B Values ◽  
Non Gaussian

scholarly journals Computing the Partial Correlation of ICA Models for Non-Gaussian Graph Signal Processing

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 22 ◽  
Author(s):  
Jordi Belda ◽  
Luis Vergara ◽  
Gonzalo Safont ◽  
Addisson Salazar
Keyword(s):  
Signal Processing ◽  
Partial Correlation ◽  
Correlation Coefficients ◽  
Real Data ◽  
Linear Estimation ◽  
Precision Matrix ◽  
Conditional Mean ◽  
Graph Signal Processing ◽  
Gaussian Case ◽  
Non Gaussian

Conventional partial correlation coefficients (PCC) were extended to the non-Gaussian case, in particular to independent component analysis (ICA) models of the observed multivariate samples. Thus, the usual methods that define the pairwise connections of a graph from the precision matrix were correspondingly extended. The basic concept involved replacing the implicit linear estimation of conventional PCC with a nonlinear estimation (conditional mean) assuming ICA. Thus, it is better eliminated the correlation between a given pair of nodes induced by the rest of nodes, and hence the specific connectivity weights can be better estimated. Some synthetic and real data examples illustrate the approach in a graph signal processing context.

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