  Consider an ergodic stationary random field $A$ on the ambient space $\mathbb
R^d$. In order to establish concentration properties for nonlinear functions
$Z(A)$, it is standard to appeal to functional inequalities like Poincar\'e or
logarithmic Sobolev inequalities in the probability space. These inequalities
are however only known to hold for a restricted class of laws (product
measures, Gaussian measures with integrable covariance, or more general Gibbs
measures with nicely behaved Hamiltonians). In this contribution, we introduce
variants of these inequalities, which we refer to as multiscale functional
inequalities and which still imply fine concentration properties, and we
develop a constructive approach to such inequalities. We consider random fields
that can be viewed as transformations of a product structure, for which the
question is reduced to devising approximate chain rules for nonlinear random
changes of variables. This approach allows us to cover most examples of random
fields arising in the modelling of heterogeneous materials in the applied
sciences, including Gaussian fields with arbitrary covariance function, Poisson
random inclusions with (unbounded) random radii, random parking and
Mat\'ern-type processes, as well as Poisson random tessellations. The obtained
multiscale functional inequalities, which we primarily develop here in view of
their application to concentration and to quantitative stochastic
homogenization, are of independent interest.
