Itô, Stratonovich, and zoom-in schemes in stochastic inflation

Abstract

The Itô and Stratonovich approaches are two ways to integrate stochastic differential equations. Detailed knowledge of the origin of the stochastic noise is needed to determine which approach suits a particular problem. I discuss this topic pedagogically in stochastic inflation, where the noise arises from a changing comoving coarse-graining scale or, equivalently, from `zooming in' into inflating space. I introduce a zoom-in scheme where deterministic evolution alternates with instantaneous zoom-in steps. I show that this alternating zoom-in scheme is equivalent to the Itô approach in the Markovian limit, while the Stratonovich approach doesn't have a similar interpretation. In the full non-Markovian setup, the difference vanishes. The framework of zoom-in schemes clarifies the relationship between computations in stochastic inflation, linear perturbation theory, and the classical $\Delta N$ formalism. It informs the numerical implementation of stochastic inflation and is a building block for a first-principles derivation of the stochastic equations.

32 pages, 4 figures. v3: Corrected typos, expanded discussion in section 4.1. Matches published version

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Cited by in corpus (5)

• Stochastic Inflation in Numerical Relativity
• Deviations from Gaussian White Noise in Stochastic Inflation
• Eternal inflation near inflection points: a challenge to primordial black hole models
• Time-reversed Stochastic Inflation
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