Harnack and HWI inequalities on infinite-dimensional spaces
The author studies dimension-free Harnack inequalities. The first to establish such a dimension-free Harnack inequality in a finite-dimensional complete Riemannian manifold was \textit{F.-Y. Wang} [Integral Equations Oper. Theory 48, No. 4, 547--552 (2004; Zbl 1074.47020)]; the fact that it is dimension-free has made it possible to investigate such inequalities also for infinite-dimensional spaces for generalised Mehler semigroups on a separable Hilbert space by \textit{M. Röckner} and \textit{F.-Y.Wang} [J. Funct. Anal. 203, No. 1, 237--261 (2003; Zbl 1059.47051)], and for an abstract Wiener space by \textit{S. Aida} and \textit{H. Kawabi} [in: L. Decreusefond (ed.) et al., Stochastic analysis and related topics VII. Boston: Birkhäuser. Prog. Probab. 48, 77--124 (2001; Zbl 0976.60077)]. The author of the present paper investigates Harnack inequalities for the heat semigroup on the based loop group as well as for the Ornstein-Uhlenbeck semigroup on the abstract Wiener space. Although being a special case of the result in [Aida and Kawabi, loc. cit.], this latter inequality is presented here with a new direct proof. Moreover, using this inequality, the author establishes an HWI inequality on the abstract Wiener space which was also studied by \textit{S. Bobkov, I. Gentil} and \textit{M. Ledoux} [J. Math. Pure Appl. (9) 80, No. 7, 669--696 (2001; Zbl 1038.35020)] using a different method.
scientific article; zbMATH DE number 5912929