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.

arXiv comment32 pages, 4 figures. v3: Corrected typos, expanded discussion in section 4.1. Matches published version

References in corpus (60)

• Planck 2018 results. X. Constraints on inflation
• A General Analytic Formula for the Spectral Index of the Density Perturbations produced during Inflation
• A new approach to the evolution of cosmological perturbations on large scales
• A general proof of the conservation of the curvature perturbation
• Cosmological perturbations
• From the Big Bang Theory to the Theory of a Stationary Universe
• Super-Horizon Scale Dynamics of Multi-Scalar Inflation
• Inflation and Eternal Inflation
• Correlation Functions in Stochastic Inflation
• Quantum diffusion during inflation and primordial black holes
• The exponential tail of inflationary fluctuations: consequences for primordial black holes
• Primordial Black Holes from Inflation and Quantum Diffusion
• Ultra slow-roll inflation demystified
• Recycling universe
• Primordial black hole formation with non-Gaussian curvature perturbations
• Revisiting non-Gaussianity from non-attractor inflation models
• Non-Gaussian tail of the curvature perturbation in stochastic ultra-slow-roll inflation: implications for primordial black hole production
• Anatomy of single-field inflationary models for primordial black holes
• Quantum diffusion beyond slow-roll: implications for primordial black-hole production
• PBH in single field inflation: the effect of shape dispersion and non-Gaussianities
• The Formation Probability of Primordial Black Holes
• The art of simulating the early Universe -- Part I
• Ultra-slow-roll inflation with quantum diffusion
• Logarithmic Duality of the Curvature Perturbation
• Stochastic inflation beyond slow roll
• A new algorithm for calculating the curvature perturbations in stochastic inflation
• Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation
• Stochastic Ultra Slow Roll Inflation
• Non-perturbative approach for curvature perturbations in stochastic-$\delta N$ formalism
• Failure of the stochastic approach to inflation beyond slow-roll
• Power spectrum in stochastic inflation
• Primordial black holes from stochastic tunnelling
• A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation
• Statistics of coarse-grained cosmological fields in stochastic inflation
• Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation
• Canonical transformations and squeezing formalism in cosmology
• Statistical nature of infrared dynamics on de Sitter background
• Stochastic constant-roll inflation and primordial black holes
• Uncertainties of predictions in models of eternal inflation
• Review on stochastic approach to inflation
• Inflation is always semi-classical: Diffusion domination overproduces Primordial Black Holes
• Stochastic inflation at all order in slow-roll parameters: foundations
• The separate-universe approach and sudden transitions during inflation
• Can all the infrared secular growth really be understood as increase of classical statistical variance?
• Martingales for physicists: A treatise on stochastic thermodynamics and beyond
• Quantum Diffusion in Sharp Transition to Non-Slow-Roll Phase
• Inflationary stochastic anomalies
• Numerical stochastic inflation constrained by frozen noise
• Multiple Field Ultra Slow Roll Inflation: Primordial Black Holes From Straight Bulk And Distorted Boundary
• Uphill inflation
• Large Deviations in the Early Universe
• Numerically modeling stochastic inflation in slow-roll and beyond
• Primordial black hole compaction function from stochastic fluctuations in ultra-slow-roll inflation
• Stochastic Inflation in General Relativity
• On the factor ordering problem in stochastic inflation
• Stochastic inflation beyond slow roll: noise modelling and importance sampling
• Quantum diffusion and large primordial perturbations from inflation
• Constant roll and non-Gaussian tail in light of logarithmic duality
• A numerical approach to stochastic inflation and primordial black holes
• Theory of Special Relativity

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
• Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function