  We develop a non-linear framework for describing long-wavelength
perturbations in multiple-field inflation. The basic variables describing
inhomogeneities are defined in a non-perturbative manner, are invariant under
changes of time slicing on large scales and include both matter and metric
perturbations. They are combinations of spatial gradients generalising the
gauge-invariant variables of linear theory. Dynamical equations are derived and
supplemented with stochastic source terms which provide the long-wavelength
initial conditions determined from short-wavelength modes. Solutions can be
readily obtained via numerical simulations or analytic perturbative expansions.
The latter are much simpler than the usual second-order perturbation theory.
Applications are given in a companion paper.
