scholarly journals BV functions in a Gelfand triple for differentiable measure and its applications

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Michael Röckner ◽  
Rongchan Zhu ◽  
Xiangchan Zhu
Keyword(s):  
Dirichlet Form ◽  
Adjoint Operator ◽  
Stochastic Quantization ◽  
Bv Functions ◽  
Form Theory ◽  
Differentiable Measure ◽  
Definition Of ◽  
Non Gaussian ◽  
Gelfand Triple ◽  
Quantization Problem

AbstractIn this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator

Dirichlet Form Theory and its Applications

2014 ◽  
Vol 11 (4) ◽  
pp. 2667-2756
Author(s):  
Sergio Albeverio ◽  
Zhen-Qing Chen ◽  
Masatoshi Fukushima ◽  
Michael Röckner
Keyword(s):  
Dirichlet Form ◽  
Form Theory

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.

A REMARK ON THE GENERATOR OF A RIGHT-CONTINUOUS MARKOV PROCESS

2007 ◽  
Vol 10 (04) ◽  
pp. 633-640 ◽  
Author(s):  
MICHAEL RÖCKNER ◽  
GERALD TRUTNAU
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Borel Measure ◽  
Initial Condition ◽  
Full Support ◽  
Finite Borel Measure ◽  
Continuous Markov Process ◽  
Form Theory ◽  
Densely Defined ◽  
Generalized Dirichlet

Given a right-continuous Markov process (Xt)t ≥ 0 on a second countable metrizable space E with transition semigroup (pt)t ≥ 0, we prove that there exists a σ-finite Borel measure μ with full support on E, and a closed and densely defined linear operator [Formula: see text] generating (pt)t ≥ 0 on Lp (E; μ). In particular, we solve the corresponding Cauchy problem in Lp (E; μ) for any initial condition [Formula: see text]. Furthermore, for any real β > 0 we show that there exists a generalized Dirichlet form which is associated to (e-βt pt)t ≥ 0. If the β-subprocess of (Xt)t ≥ 0 corresponding to (e-βt pt)t ≥ 0, β > 0, is μ-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for [Formula: see text]. If (Xt)t ≥ 0 is transient, then β can be chosen to be zero.

Definition of a Complex Domain Adjoint for Use in Optimal Control Problems

1978 ◽  
Vol 11 (1) ◽  
pp. 33-35
Author(s):  
M. J. Grimble
Keyword(s):  
Linear Time ◽  
Adjoint Operator ◽  
Adjoint System ◽  
Simple Relationship ◽  
Complex Frequency ◽  
Simple Method ◽  
Linear Time Invariant ◽  
Definition Of ◽  
The Time Domain ◽  
Linear Time Invariant System

A complex frequency version of the time domain adjoint operator for a linear time-invariant system is obtained. A very simple relationship is shown to exist between this operator and the system transfer function matrix. A simple method of simulating the adjoint system is described.

Spectral Multipliers for Laplacians Associated with some Dirichlet Forms

2008 ◽  
Vol 51 (3) ◽  
pp. 581-607 ◽  
Author(s):  
Andrea Carbonaro ◽  
Giancarlo Mauceri ◽  
Stefano Meda
Keyword(s):  
Lebesgue Measure ◽  
Essential Spectrum ◽  
Dirichlet Form ◽  
Adjoint Operator ◽  
Dirichlet Forms ◽  
The Self ◽  
Spectral Multipliers ◽  
Curvature Condition ◽  
Image Position ◽  
Conditions At Infinity

AbstractLet be the self-adjoint operator associated with the Dirichlet formwhere ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

scholarly journals ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

2017 ◽  
Vol 23 (2) ◽  
pp. 32-43
Author(s):  
M. V. Kukushkin
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Fractional Differentiation ◽  
Qualitative Properties ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Integral Equality ◽  
Definition Of

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

scholarly journals A localization formula in Dirichlet form theory

2012 ◽  
Vol 140 (5) ◽  
pp. 1815-1822
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Localization Formula ◽  
Form Theory

BV functions in a Gelfand triple and the stochastic reflection problem on a convex set of a Hilbert space

2010 ◽  
Vol 348 (21-22) ◽  
pp. 1175-1178
Author(s):  
Michael Röckner ◽  
Rongchan Zhu ◽  
Xiangchan Zhu
Keyword(s):  
Hilbert Space ◽  
Convex Set ◽  
Bv Functions ◽  
Gelfand Triple

Reflected Dirichlet Spaces

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Harmonic Functions ◽  
Dirichlet Space ◽  
Random Variables ◽  
Finite Energy ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Hunt Process ◽  
Equivalent Definition ◽  
Definition Of

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. The chapter next establishes several equivalent notions of reflected Dirichlet space (ℰ ref,ℱ ref) for a regular transient Dirichlet form (E,F). One of these equivalent notions is then used to define reflected Dirichlet space for a regular recurrent Dirichlet form. Moreover, the chapter gives yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms. Various concrete examples of reflected Dirichlet spaces are also exhibited for regular Dirichlet forms. Finally, the chapter defines a Silverstein extension of a quasi-regular Dirichlet form (E,F) on L²(E; m) and investigates the equivalence of analytic and probabilistic concepts of harmonicity.

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