The Notion of Convexity and Concavity on Wiener Space

Abstract

We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting. Academic Press

An Estimate of the Gap of Spectrum of Schrödinger Operators which Generate Hyperbounded Semigroups

“…Then an LSI and Poincare 's inequality hold on X i . This is proved in Example in p. 422 in [18]. For the completeness, we will give a proof.…”

A new approach to Poincaré-type inequalities on the Wiener space

“…The previous inequality was adapted to the context of abstract Wiener spaces by Feyel andÜstünel in [6]. The aim of the present paper is to prove a new type of Poincaré inequality for a class of probability measures on an abstract Wiener space which is not included in the family of log-concave measures.…”

“…Let F be of the form p(δ(h 1 ), ..., δ(h n )) where p : R n → R is a polynomial and h 1 , ..., h n ∈ H. Functions of this type belong to G and they are dense in L p (W, µ) and D k,p for any p ≥ 1 and k ∈ N. Then we can write 6) where for any j ≥ 1, {e…”

“…Moreover, the family of probability measures for which our Poincaré inequality is proved, namely measures of the type νdµ with ν strongly positive, is in general not included in the class of log-concave measures, which is the class of probability measure considered by Brascamp and Lieb [2] (see the example at the end of Section 3) and Feyel andÜstünel [6] (in the abstract Wiener space setting). …”

“…This space is well defined through the closability of the gradient operator with respect to the norm in L 2 (W, νdµ) which is however not known in the general case. This is why for instance the Poincaré inequality for log-concave measures on abstract Wiener spaces obtained in [6] is proved only for cilindrical smooth functions.…”