Weighted functional inequalities: Constructive approach

Mitia Duerinckx (1, 2) , Antoine Gloria (1)

1 ULB - Département de Mathématique [Bruxelles]
2 MEPHYSTO - Quantitative methods for stochastic models in physics

Consider an ergodic stationary random field A on the ambient space R d. In a companion article we introduced the notion of weighted functional inequalities, which extend standard functional inequalities like spectral gap, covariance, and logarithmic Sobolev inequalities, and we studied the associated concentration properties for nonlin-ear functions X(A) of the field. In the present contribution we develop a constructive approach to produce random fields that satisfy such weighted functional inequalities. The construction typically relies on devising approximate chain rules for nonlinear and random changes of variables for random fields. This approach allows us to cover Gauss-ian fields with non-necessarily integrable covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations (Voronoi or Delaunay). These weighted functional inequalities , which we primarily develop here in view of their application to quantitative stochastic homogenization, are of independent interest.

Domaines: Probabilités [math.PR]

Mitia Duerinckx, Antoine Gloria. Weighted functional inequalities: Constructive approach. 2017. ⟨hal-01633041v1⟩