Classical Dirichlet forms on topological vector spaces --- closability and a Cameron-Martin formula
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) in such areas. An exhaustive bibliography of almost 60 classical and modern references is provided. A note added in proof contains many valuable informations on the recent progress concerning Dirichlet forms on topological vector spaces and their further applications.
Dirichlet forms on topological vector spaces
scientific article; zbMATH DE number 4707
scientific article