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Stochastic inflation is an effective theory describing the super-Hubble,coarse-grained, scalar fields driving inflation, by a set of Langevinequations. We previously highlighted the difficulty of deriving a theory ofstochastic inflation that is invariant under field redefinitions, and the linkwith the ambiguity of discretisation schemes defining stochastic differentialequations. In this paper, we solve the issue of these "inflationary stochasticanomalies" by using the Stratonovich discretisation satisfying generalcovariance, and identifying that the quantum nature of the fluctuating fieldsentails the existence of a preferred frame defining independent stochasticnoises. Moreover, we derive physically equivalent It\^o-Langevin equations thatare manifestly covariant and well suited for numerical computations. Theseequations are formulated in the general context of multifield inflation withcurved field space, taking into account the coupling to gravity as well as thefull phase space in the Hamiltonian language, but this resolution is alsorelevant in simpler single-field setups. We also develop a path-integralderivation of these equations, which solves conceptual issues of the heuristicapproach made at the level of the classical equations of motion, and allows inprinciple to compute corrections to the stochastic formalism. Using theSchwinger-Keldysh formalism, we integrate out small-scale fluctuations, derivethe influence action that describes their effects on the coarse-grained fields,and show how the resulting coarse-grained effective Hamiltonian action can beinterpreted to derive Langevin equations with manifestly real noises. Althoughthe corresponding dynamics is not rigorously Markovian, we show the covariant,phase-space Fokker-Planck equation for the Probability Density Function offields and momenta when the Markovian approximation is relevant [...]

A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation