Itô, Stratonovich, and zoom-in schemes in stochastic inflation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912342420226048 |
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| author | Tomberg, Eemeli |
| author_facet | Tomberg, Eemeli |
| contents | 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 $ΔN$ formalism. It informs the numerical implementation of stochastic inflation and is a building block for a first-principles derivation of the stochastic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Itô, Stratonovich, and zoom-in schemes in stochastic inflation Tomberg, Eemeli Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Theory 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 $ΔN$ formalism. It informs the numerical implementation of stochastic inflation and is a building block for a first-principles derivation of the stochastic equations. |
| title | Itô, Stratonovich, and zoom-in schemes in stochastic inflation |
| topic | Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2411.12465 |