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Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space. (English) Zbl 1075.60059
Kunita, Hiroshi (ed.) et al., Stochastic analysis and related topics in Kyoto. In honour of Kiyoshi Itô. Lectures given at the conference, Kyoto, Japan, September 4–7, 2002. Tokyo: Mathematical Society of Japan (ISBN 4-931469-26-4/hbk). Advanced Studies in Pure Mathematics 41, 121-140 (2004).
If $$X$$ is a compact Hausdorff space, then the Riesz representation theorem says that any bounded linear functional $$T$$ on $$C(X)$$ can be expressed as $$T(v)= \int_X v(x) d\nu(x)$$ for some finite signed measure $$\nu$$. A similar result holds for $$T(v)={\mathcal E}(u,v)$$ for a quasi-regular Dirichlet form $$({\mathcal E},{\mathcal F})$$. The main purpose of this paper is to generalize this result to the case of vector lattice by an analytic method. Let $$X$$ be a topological space and $$\lambda$$ a Borel measure on $$X$$. Consider a separable uniformly convex Banach space $$(D,\| \cdot\| _D)$$ constituting from a subspace of Borel measurable functions on $$X$$. Further assume that any sequence converging to 0 in $$D$$ contains a subsequence which converges to 0 $$\lambda$$-a.s. Similarly to the case of Dirichlet forms, assume that $$D$$ satisfies the normal contraction and quasi-regularity properties. Choosing suitable functions $$\rho \in D$$ and $$\xi$$ on $$[0,\infty)$$, a capacity related to $$D$$ is introduced. For this capacity, the notion of nest is defined as in the case of quasi-regular Dirichlet forms. Then the main theorem of this paper says that the following two statements are equivalent. (i) There exists a nest $$\{F_k\}$$ and constants $$\{C_k\}$$ such that $$T(v) \leq C_k \| v\| _{L^\infty(X)}, \;\forall v \in D_{b,F_k}$$. (ii) There exists a finite signed smooth measure $$\nu$$ and a nest $$\{F_f'\}$$ such that $$T(v)=\int_X \tilde{v}(z)\nu(dz), \;\forall v \in D_{b,F_k'}$$, where $$D_A$$ is the family of elements of $$D$$ which vanishes on $$X\setminus A$$ a.e. This result is applied to a function of bounded variation on Wiener space to improve the proof of the smoothness of the associated measure. Furthermore a nontrivial example of function of bounded variation is also given.
For the entire collection see [Zbl 1050.60002].

MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 46B42 Banach lattices 60J45 Probabilistic potential theory