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

We prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures that are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [22]) of their Radon–Nikodym densities. In general, measures of this type do not belong to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a pointwise identity relating Wick and ordinary products and on the notion of strong positivity which is connected to the non-negativity of Wick powers. Our technique also leads to a partial generalization of the Houdré and Kagan [11] and Houdré and Pérez-Abreu [12] Poincaré-type inequalities.

The Gaussian Radon transform and machine learning

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform.

AN INVERSION FORMULA FOR THE GAUSSIAN RADON TRANSFORM FOR BANACH SPACES

We prove a disintegration theorem for the Gaussian Radon transform Gf on Banach spaces and use the Segal–Bargmann transform on abstract Wiener spaces to find a procedure to obtain f from its Gaussian Radon transform Gf.