  Malliavin calculus provides a characterization of the centered model in
regularity structures that is stable under removing the small-scale cut-off. In
conjunction with a spectral gap inequality, it yields the stochastic estimates
of the model.
  This becomes transparent on the level of a notion of model that parameterizes
the solution manifold, and thus is indexed by multi-indices rather than trees,
and which allows for a more geometric than combinatorial perspective. In these
lecture notes, this is carried out for a PDE with heat operator, a cubic
nonlinearity, and driven by additive noise, reminiscent of the stochastic
quantization of the Euclidean $\phi^4$ model.
  More precisely, we informally motivate our notion of the model $(\Pi,\Gamma)$
as charts and transition maps, respectively, of the nonlinear solution
manifold. These geometric objects are algebrized in terms of formal power
series, and their algebra automorphisms. We will assimilate the directional
Malliavin derivative to a tangent vector of the solution manifold. This means
that it can be treated as a modelled distribution, thereby connecting
stochastic model estimates to pathwise solution theory, with its analytic tools
of reconstruction and integration. We unroll an inductive calculus that in an
automated way applies to the full subcritical regime.
