Stochastic inflation is an effective theory describing the super-Hubble, coarse-grained, scalar fields driving inflation, by a set of Langevin equations. We previously highlighted the difficulty of deriving a theory of stochastic inflation that is invariant under field redefinitions, and the link with the ambiguity of discretisation schemes defining stochastic differential equations. In this paper, we solve the issue of these "inflationary stochastic anomalies" by using the Stratonovich discretisation satisfying general covariance, and identifying that the quantum nature of the fluctuating fields entails the existence of a preferred frame defining independent stochastic noises. Moreover, we derive physically equivalent It-Langevin equations that are manifestly covariant and well suited for numerical computations. These equations are formulated in the general context of multifield inflation with curved field space, taking into account the coupling to gravity as well as the full phase space in the Hamiltonian language, but this resolution is also relevant in simpler single-field setups. We also develop a path-integral derivation of these equations, which solves conceptual issues of the heuristic approach made at the level of the classical equations of motion, and allows in principle to compute corrections to the stochastic formalism. Using the Schwinger-Keldysh formalism, we integrate out small-scale fluctuations, derive the influence action that describes their effects on the coarse-grained fields, and show how the resulting coarse-grained effective Hamiltonian action can be interpreted to derive Langevin equations with manifestly real noises. Although the corresponding dynamics is not rigorously Markovian, we show the covariant, phase-space Fokker-Planck equation for the Probability Density Function of fields and momenta when the Markovian approximation is relevant, and we give analytical approximations for the noises' amplitudes in multifield scenarios.

随机暴胀是一种有效的理论，通过一组朗之万方程描述了超哈勃、粗粒度、标量场驱动暴胀。我们之前强调了推导出在场重新定义下不变的随机膨胀理论的困难，以及与定义随机微分方程的离散化方案的模糊性的联系。在本文中，我们通过使用满足一般协方差的 Stratonovich 离散化来解决这些“通货膨胀随机异常”的问题，并确定波动场的量子性质需要存在定义独立随机噪声的首选框架。此外，我们推导出了物理等效的 It-Langevin 方程，这些方程显然是协变的，非常适合数值计算。这些方程是在具有弯曲场空间的多场膨胀的一般背景下制定的，考虑到重力耦合以及哈密顿语言中的全相空间，但该分辨率也与更简单的单场设置相关。我们还开发了这些方程的路径积分推导，它解决了在经典运动方程水平上进行的启发式方法的概念问题，并且原则上允许计算对随机形式主义的修正。使用 Schwinger-Keldysh 形式主义，我们整合了小尺度波动，推导出描述它们对粗粒度场影响的影响作用，并展示如何解释由此产生的粗粒度有效哈密顿量作用以推导出朗之万方程真正的噪音。