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Aims. This series of papers aims at building a new formalism specificallytailored to study the impact of turbulence on the global modes of oscillationin solar-like stars. This first paper aims at deriving a linear wave equationthat directly and consistently contains the turbulence as an input to themodel, and therefore naturally contains the information on the coupling betweenthe turbulence and the modes, through the stochasticity of the equations.  Methods. We use a Lagrangian stochastic model of turbulence based onProbability Density Function methods to describe the evolution of theproperties of individual fluid particles through stochastic differentialequations. We then transcribe these stochastic differential equations from aLagrangian frame to an Eulerian frame, more adapted to the analysis of stellaroscillations. We combine this method with Smoothed Particle Hydrodynamics,where all the mean fields appearing in the Lagrangian stochastic model areestimated directly from the set of fluid particles themselves, through the useof a weighting kernel function allowing to filter the particles present in anygiven vicinity. The resulting stochastic differential equations on Eulerianvariables are then linearised.  Results. We obtain a stochastic, linear wave equation governing the timeevolution of the relevant wave variables, while at the same time containing theeffect of turbulence. The wave equation generalises the classical, unperturbedpropagation of acoustic waves in a stratified medium to a form that, byconstruction, accounts for the impact of turbulence on the mode in a consistentway. The effect of turbulence consists in a non-homogeneous forcing term,responsible for the stochastic driving of the mode, and a stochasticperturbation to the homogeneous part of the wave equation, responsible for boththe damping of the mode and the modal surface effects.

Coupling between turbulence and solar-like oscillations: a combined Lagrangian PDF/SPH approach. I -- The stochastic wave equation