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Representation formula for the entropy and functional inequalities

Joseph Lehec (2013)

Annales de l'I.H.P. Probabilités et statistiques

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell’s formula for the Laplace transform. As an application, we give simple proofs of a number of functional inequalities.

Representation formulae for (C₀) m-parameter operator semigroups

Mi Zhou, George A. Anastassiou (1996)

Annales Polonici Mathematici

Some general representation formulae for (C₀) m-parameter operator semigroups with rates of convergence are obtained by the probabilistic approach and multiplier enlargement method. These cover all known representation formulae for (C₀) one- and m-parameter operator semigroups as special cases. When we consider special semigroups we recover well-known convergence theorems for multivariate approximation operators.

Representation of Itô integrals by Lebesgue/Bochner integrals

Qi Lü, Jiongmin Yong, Xu Zhang (2012)

Journal of the European Mathematical Society

In [Yong 2004], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later...

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