Classical Dirichlet forms on topological vector spaces --- closability and a Cameron-Martin formula
type: Item
instance of: scholarly article
title: Classical Dirichlet forms on topological vector spaces --- closability and a Cameron-Martin formula
zbMATH Open document ID: 0737.46036
DOI: 10.1016/0022-1236(90)90113-Y
published in: Journal of Functional Analysis
publication date: 1992-06-25T00:00:00Z
review text: Let \(E\) be a locally convex Hausdorff topological vector space, and \(\mu\) be a finite positive measure on \(E\). Classical Dirichlet forms are closures in \(L_ 2(E;\mu)\) of the linear span of differential forms \[ {\mathcal E}_ k(u,v)=\int_ E \partial u/\partial k \cdot\partial v/\partial k d\mu, \] where \(u,v\) are either \(C_ 0^ \infty\)-functions in the finite dimensional case, or bounded smooth functions that are subject to certain additional constraints, when \(E\) is infinite dimensional, \(k\in E\), \(k\neq 0\), and the directional derivative along \(k\) is involved. The question of closability, which is an invariant property under linear operations, of forms \({\mathcal E}_ k(u,v)\), has rised from the one-dimensional classical context, and its multidimensional counterpart [cf. \textit{M. Fukushima}, On a stochastic calculus related to Dirichlet forms and distorted Brownian Motion, Physical Reports 77, 255-262 (1981)]. In particular, the affirmative answer to the conjecture, that the closability along all basis vectors \(k\) in a finite dimensional Euclidean space implies the absolute continuity of \(\mu\) with respect to the Lebesgue measure, follows from the main result of this paper. The theorem in Section 3 gives necessary and sufficient conditions for forms \({\mathcal E}_ k\) to be well defined and closable, and describes the Dirichlet forms explicitly. In addition, the corresponding Radon-Nikodym derivatives are characterized. The case of great importance, i.e., when \(\mu\) is quasi-invariant with respect to \(k\), is investigated in Section 4, yielding a Cameron-Martin- type formula. The classical results are given new or modified proofs (Section 2). There are strong connections with the theory of finite dimensional quantum fields, theory of Markov processes and their generators, finite- and infinite dimensional stochastic differential equations, diffusions on manifolds, etc. The study is both motivated (Section 0) and applicable (Section 5) i…
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 47N50
zbMATH DE Number: 4707
zbMATH Keywords: Malliavin's calculus
zbMATH Keywords: Brownian motion
zbMATH Keywords: closability
zbMATH Keywords: Radon-Nikodym derivatives
zbMATH Keywords: Cameron-Martin-type formula
zbMATH Keywords: finite dimensional quantum fields
zbMATH Keywords: Markov processes
zbMATH Keywords: Dirichlet forms on topological vector spaces
author: S. Albeverio
author: Michael Röckner
reviewed by: Jerzy Szulga
MaRDI profile type: Publication
cites work: Q3742397
cites work: Capacity and quantum mechanical tunneling
cites work: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time
cites work: Dirichlet forms and diffusion processes on rigged Hilbert spaces
cites work: Q4160194
cites work: Energy forms, Hamiltonians, and distorted Brownian paths
cites work: Q3139601
cites work: Q3033068
cites work: Classical Dirichlet forms on topological vector spaces - the construction of the associated diffusion process
cites work: Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques
cites work: Espaces de Dirichlet. I: Le cas élémentaire
cites work: DIRICHLET SPACES
cites work: Stochastic quantization of field theory in finite and infinite volume
cites work: Q3784937
cites work: Formes de Dirichlet générales et densité des variables aléatoires réelles sur l'espace de Wiener. (General Dirichlet forms and density of real random variables on Wiener space)
cites work: Transformations of Wiener integrals under translations
cites work: Measurable norms and some Banach space valued Gaussian processes
cites work: Q3957682
cites work: Q5565478
cites work: Schwinger functions and their generating functionals. II. Markovian and generalized path space measures on \(\mathcal S'\)
cites work: Q3855442
cites work: Basic properties of Brownian motion and a capacity on the Wiener space
cites work: Q3910965
cites work: Q5581156
cites work: Logarithmic Sobolev Inequalities
cites work: The \(P(\phi)_2\) Euclidean quantum field theory as classical statistical mechanics. II.
cites work: Q5106442
cites work: Quadratic functionals of Brownian motion
cites work: Q4202615
cites work: Dirichlet forms and white noise analysis
cites work: On the stochastic quantization of field theory
cites work: Calcul d'intégrales et de dérivées en dimension infinie
cites work: Gaussian measures in Banach spaces
cites work: Q3959912
cites work: Q4197841
cites work: Q5679204
cites work: The free Markoff field
cites work: The disintegration of invariant measures
cites work: Specifications and Martin boundaries for \(\mathcal P(\phi)_2\)-random fields
cites work: Traces of harmonic functions and a new path space for the free quantum field
cites work: Q4204923
cites work: Q3692031
cites work: Q4052400
cites work: Markov processes with creation of particles
cites work: Q3643263
cites work: On the uniqueness of Markovian self-adjoint extension of diffusion operators on infinite dimensional spaces
cites work: Theory of operator algebras I.
cites work: Lectures on stochastic differential equations and Malliavin calculus
full work available at URL: 0022-1236(90)90113-y
OpenAlex ID: W1965805464
description: scientific article; zbMATH DE number 4707