pith. sign in

arxiv: 2605.21929 · v1 · pith:H5XO4NEAnew · submitted 2026-05-21 · ✦ hep-th · gr-qc

Stochastic inflation as an open quantum system II: open effective field theory and stochastic matching

Yue-Zhou Li This is my paper

Pith reviewed 2026-05-22 05:37 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords stochastic inflationopen quantum systemeffective field theoryrenormalization groupWilson kernelsFokker-Planck equationKlein-Kramers equationinflationary cosmology
0
0 comments X

The pith

Stochastic inflation as an open quantum system needs a distinct stochastic renormalization channel with nonlocal Wilson kernels beyond Gaussian order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an open effective field theory for the reduced density matrix of long-wavelength modes in stochastic inflation. It shows that this theory has two renormalization group flows: the usual Wilsonian one and a new stochastic channel due to the system's openness. Focusing on the stochastic channel using a hard cutoff, Gaussian and non-Gaussian diffusion appear as effective operators in the influence functional. These must match the correlators and form factors of the full perturbative theory using a method of regions in time. Beyond the Gaussian level, the matching involves nonlocal and non-Markovian Wilson kernels instead of local coefficients.

Core claim

The open effective field theory for stochastic inflation possesses a stochastic renormalization channel isolated in the hard cutoff scheme. In this channel, effective operators for Gaussian and non-Gaussian diffusion are identified by matching to the full theory, and beyond Gaussian order the matching data consist of nonlocal non-Markovian Wilson kernels. This leads to a bare Hamiltonian density and nonlocal functional master equations such as the Fokker-Planck equation for the diagonal density matrix and the Klein-Kramers equation for the Wigner functional.

What carries the argument

The stochastic renormalization channel in the hard cutoff scheme that generates nonlocal Wilson kernels for the influence functional through time-region matching.

Load-bearing premise

The hard-cutoff scheme cleanly isolates a stochastic RG channel whose matching to the full theory can be performed independently of the conventional Wilsonian flow.

What would settle it

Computing the next-to-Gaussian correlators or form factors in the full theory and finding that they cannot be matched by any choice of nonlocal kernels in the open EFT would disprove the claim.

read the original abstract

We further develop the proposal in Phys.\ Rev.\ Lett.\ \textbf{136} 071501 that interprets stochastic inflation as an open quantum system, by constructing the open effective field theory for the reduced density matrix of long wavelength modes. We clarify that this open effective field theory enjoys two renormalization group flows: the conventional Wilsonian channel, and a stochastic channel arising from the openness that has no counterpart in ordinary Wilsonian effective field theory. Focusing on the stochastic channel in the hard cutoff scheme, we identify both Gaussian and non-Gaussian diffusion as effective operators in the influence functional, and show that they are required by matching onto correlators and form factors of the perturbative full theory through a method-of-region in time. Beyond Gaussian order, the matching data are no longer local Wilson coefficients but nonlocal and non-Markovian Wilson kernels. We then obtain the bare Hamiltonian density of this open effective field theory and derive its nonlocal functional master equations, including the Fokker-Planck equation for the diagonal density matrix and the Klein-Kramers equation for the Wigner functional, with their zero-modess simplifications discussed. Finally, we take a first step toward a continuum version of this open effective field theory, replacing the hard cutoff by an analytic regulator in the stochastic channel, and demonstrate stochastic renormalization using a massive scalar as an example.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. This paper extends the open quantum system interpretation of stochastic inflation by constructing the open effective field theory for the reduced density matrix of long-wavelength modes. It identifies two renormalization group flows: the standard Wilsonian channel and a novel stochastic channel without analog in ordinary EFTs. In the hard cutoff scheme, Gaussian and non-Gaussian diffusion operators in the influence functional are matched to full-theory correlators and form factors via time method-of-regions. Beyond Gaussian order, matching produces nonlocal non-Markovian Wilson kernels. The bare Hamiltonian is obtained, leading to nonlocal functional master equations (Fokker-Planck for diagonal density matrix, Klein-Kramers for Wigner functional). A continuum version with analytic regulator is explored, with stochastic renormalization demonstrated for a massive scalar.

Significance. Should the central claims on the clean separation of the stochastic channel and the validity of the nonlocal kernel matching hold, the work offers a promising extension of EFT techniques to stochastic inflation, enabling systematic treatment of non-Gaussian effects through open-system methods. The derivation of functional master equations and the initial continuum limit provide concrete tools for further development. This could impact calculations of primordial non-Gaussianity and related observables in inflationary cosmology.

major comments (2)
  1. [Abstract] Abstract, paragraph on stochastic channel in hard cutoff scheme: the manuscript asserts that the hard-cutoff scheme cleanly isolates the stochastic RG channel, permitting independent matching onto full-theory quantities independent of the conventional Wilsonian flow. However, for non-Gaussian operators, the time integrals in the influence functional that generate the kernels may include overlapping contributions from modes near the cutoff. This raises the possibility that the extracted nonlocal kernels contain pieces that would be absorbed into the Wilsonian flow in a standard treatment. An explicit demonstration that the two channels remain orthogonal when non-Gaussian diffusion terms are included is required to substantiate the claim that the matching data are purely stochastic in origin.
  2. [Section on functional master equations] The derivation of the nonlocal functional master equations (Fokker-Planck and Klein-Kramers) follows the matching procedure. To confirm consistency, an explicit check should be provided that the non-Markovian kernels preserve the required properties of the density matrix (e.g., trace preservation and positivity) when acting on the zero-mode sector, particularly for the non-Gaussian terms.
minor comments (1)
  1. The notation distinguishing local Wilson coefficients from nonlocal Wilson kernels could be introduced with a clear definition or table early in the text to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph on stochastic channel in hard cutoff scheme: the manuscript asserts that the hard-cutoff scheme cleanly isolates the stochastic RG channel, permitting independent matching onto full-theory quantities independent of the conventional Wilsonian flow. However, for non-Gaussian operators, the time integrals in the influence functional that generate the kernels may include overlapping contributions from modes near the cutoff. This raises the possibility that the extracted nonlocal kernels contain pieces that would be absorbed into the Wilsonian flow in a standard treatment. An explicit demonstration that the two channels remain orthogonal when non-Gaussian diffusion terms are included is required to substantiate the claim that the matching data are purely stochastic in origin.

    Authors: We agree that an explicit demonstration of orthogonality for the non-Gaussian case would clarify the separation. Our matching procedure employs the time method-of-regions to isolate secular, long-time contributions in the influence functional, which define the stochastic channel, from short-time pieces associated with Wilsonian renormalization. To directly address the concern, we will add an explicit decomposition of the relevant time integrals for the non-Gaussian diffusion operators in the revised manuscript, confirming that the extracted nonlocal kernels contain no local Wilsonian counterterms. revision: yes

  2. Referee: [Section on functional master equations] The derivation of the nonlocal functional master equations (Fokker-Planck and Klein-Kramers) follows the matching procedure. To confirm consistency, an explicit check should be provided that the non-Markovian kernels preserve the required properties of the density matrix (e.g., trace preservation and positivity) when acting on the zero-mode sector, particularly for the non-Gaussian terms.

    Authors: We thank the referee for highlighting the importance of this consistency check. The master equations are obtained from the influence functional of the underlying unitary theory, which ensures trace preservation by construction. For positivity, we will add an explicit verification in the revised manuscript for the zero-mode sector, acting with the leading non-Gaussian non-Markovian kernels on representative states and confirming preservation within the regime of validity of the effective description. revision: yes

Circularity Check

0 steps flagged

No significant circularity: matching to external perturbative correlators keeps derivation independent

full rationale

The paper develops an open EFT by identifying Gaussian and non-Gaussian diffusion operators in the influence functional and matching them onto correlators and form factors of the perturbative full theory via method-of-region in time. This matching supplies the nonlocal non-Markovian Wilson kernels beyond Gaussian order. The stochastic renormalization channel is presented as arising from the openness of the system and is treated as separable in the hard-cutoff scheme. No equation or step in the provided derivation chain reduces a claimed prediction or kernel to a fitted parameter or self-citation by construction. The cited PRL proposal supplies the initial open-system interpretation but is not invoked as a uniqueness theorem or load-bearing justification for the specific matching results or master equations derived here. The construction therefore remains self-contained against the external full-theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the open-quantum-system interpretation of stochastic inflation and on the existence of a cleanly separable stochastic renormalization channel in the hard-cutoff scheme; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Stochastic inflation admits an open-quantum-system description in which short modes act as an environment for long modes
    Stated as the starting proposal being further developed; invoked throughout the abstract.
  • domain assumption The stochastic RG channel can be isolated and matched independently of the Wilsonian channel
    Explicitly focused on in the hard-cutoff scheme paragraph.

pith-pipeline@v0.9.0 · 5763 in / 1321 out tokens · 37767 ms · 2026-05-22T05:37:13.049162+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Focusing on the stochastic channel in the hard cutoff scheme, we identify both Gaussian and non-Gaussian diffusion as effective operators in the influence functional... Beyond Gaussian order, the matching data are no longer local Wilson coefficients but nonlocal and non-Markovian Wilson kernels.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

154 extracted references · 154 canonical work pages · 55 internal anchors

  1. [1]

    C. W. Bauer, S. Fleming and M. E. Luke,Summing Sudakov logarithms inB→X sγin effective field theory.,Phys. Rev. D63(2000) 014006, [hep-ph/0005275]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  2. [2]

    C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart,An Effective field theory for collinear and soft gluons: Heavy to light decays,Phys. Rev. D63(2001) 114020, [hep-ph/0011336]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  3. [3]

    Becher, A

    T. Becher, A. Broggio, A. Ferroglia et al.,Introduction to soft-collinear effective theory, vol. 10. Springer, 2015

    work page 2015

  4. [4]

    I. Z. Rothstein and I. W. Stewart,An Effective Field Theory for Forward Scattering and Factorization Violation,JHEP08(2016) 025, [1601.04695]

    work page arXiv 2016

  5. [5]

    R. P. Feynman and F. L. Vernon Jr,The theory of a general quantum system interacting with a linear dissipative system,Annals of physics24(1963) 118–173

    work page 1963

  6. [6]

    Schwinger,Brownian motion of a quantum oscillator,Journal of Mathematical Physics2 (1961) 407–432

    J. Schwinger,Brownian motion of a quantum oscillator,Journal of Mathematical Physics2 (1961) 407–432

    work page 1961

  7. [7]

    L. V. Keldysh,Diagram technique for nonequilibrium processes, inSelected Papers of Leonid V Keldysh, pp. 47–55. World Scientific, 2024

    work page 2024

  8. [8]

    Effective field theory of dissipative fluids

    M. Crossley, P. Glorioso and H. Liu,Effective field theory of dissipative fluids,JHEP09 (2017) 095, [1511.03646]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  9. [9]

    Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics

    H. Liu and P. Glorioso,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,PoSTASI2017(2018) 008, [1805.09331]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    W. D. Goldberger and A. Ross,Gravitational radiative corrections from effective field theory, Phys. Rev. D81(2010) 124015, [0912.4254]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  11. [11]

    M. V. S. Saketh, Z. Zhou and M. M. Ivanov,Dynamical tidal response of Kerr black holes from scattering amplitudes,Phys. Rev. D109(2024) 064058, [2307.10391]. – 59 –

    work page arXiv 2024

  12. [12]

    M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez and Z. Zhou,Gravitational Raman Scattering in Effective Field Theory: A Scalar Tidal Matching at O(G3),Phys. Rev. Lett.132(2024) 131401, [2401.08752]

    work page arXiv 2024

  13. [13]

    Glazer, A

    D. Glazer, A. Joyce, M. J. Rodriguez, L. Santoni, A. R. Solomon and L. F. Temoche, Higher-Dimensional Black Holes and Effective Field Theory,2412.21090

    work page arXiv

  14. [14]

    Caron-Huot, M

    S. Caron-Huot, M. Correia, G. Isabella and M. Solon,Gravitational Wave Scattering via the Born Series: Scalar Tidal Matching toO(G 7)and Beyond,2503.13593

    work page arXiv

  15. [15]

    M. M. Ivanov, Y.-Z. Li, J. Parra-Martinez and Z. Zhou,Gravitational Raman Scattering: a Systematic Toolkit for Tidal Effects in General Relativity,2602.06951

    work page arXiv

  16. [16]

    Dissipative effects in the Effective Field Theory of Inflation

    D. Lopez Nacir, R. A. Porto, L. Senatore and M. Zaldarriaga,Dissipative effects in the Effective Field Theory of Inflation,JHEP01(2012) 075, [1109.4192]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  17. [17]

    C. P. Burgess, R. Holman, G. Tasinato and M. Williams,EFT Beyond the Horizon: Stochastic Inflation and How Primordial Quantum Fluctuations Go Classical,JHEP03(2015) 090, [1408.5002]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  18. [18]

    S. A. Salcedo, T. Colas and E. Pajer,The open effective field theory of inflation,JHEP10 (2024) 248, [2404.15416]

    work page arXiv 2024

  19. [19]

    Colas,Open Effective Field Theories for cosmology, in58th Rencontres de Moriond on Cosmology, 5, 2024.2405.09639

    T. Colas,Open Effective Field Theories for cosmology, in58th Rencontres de Moriond on Cosmology, 5, 2024.2405.09639

    work page arXiv 2024

  20. [20]

    S. A. Salcedo, T. Colas, L. Dufner and E. Pajer,An Open System Approach to Gravity, 2507.03103

    work page arXiv

  21. [21]

    Colas,Lectures on Open Effective Field Theories, 9, 2025.2510.00140

    T. Colas,Lectures on Open Effective Field Theories, 9, 2025.2510.00140

    work page arXiv 2025

  22. [22]

    Colas, Z

    T. Colas, Z. Qin and X. Tong,Open Effective Field Theory and the Physics of Cosmological Collider Signals,2512.07941

    work page arXiv

  23. [23]

    S. A. Salcedo, T. Colas, L. Dufner and E. Pajer,Phenomenology of an Open Effective Field Theory of Dark Energy,2603.12321

    work page internal anchor Pith review Pith/arXiv arXiv

  24. [24]

    S. A. Salcedo, T. Colas, P. Suman, B. Zhang, J. Fergusson and E. P. S. Shellard,Primordial non-Gaussianity constraints on dissipative inflation,2603.13473

    work page arXiv

  25. [25]

    Cespedes, Z

    S. Cespedes, Z. Qin and D.-G. Wang,λϕ 4 as an effective theory in de Sitter,JHEP05(2026) 143, [2510.25826]

    work page arXiv 2026

  26. [26]

    A. A. Starobinsky,STOCHASTIC DE SITTER (INFLATIONARY) STAGE IN THE EARLY UNIVERSE,Lect. Notes Phys.246(1986) 107–126

    work page 1986

  27. [27]

    V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger,Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part

    work page

  28. [28]

    Rept.215(1992) 203–333

    Extensions,Phys. Rept.215(1992) 203–333

    work page 1992

  29. [29]

    J. M. Maldacena,Non-Gaussian features of primordial fluctuations in single field inflationary models,JHEP05(2003) 013, [astro-ph/0210603]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  30. [30]

    Second-Order Cosmological Perturbations from Inflation

    V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto,Second order cosmological perturbations from inflation,Nucl. Phys. B667(2003) 119–148, [astro-ph/0209156]. – 60 –

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  31. [31]

    Cosmological perturbation theory

    R. Durrer,Cosmological perturbation theory,Lect. Notes Phys.653(2004) 31–70, [astro-ph/0402129]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  32. [32]

    Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp

    D. Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp. 523–686, 2011.0907.5424. DOI

    work page arXiv 2011

  33. [33]

    Wang,Inflation, cosmic perturbations and non-gaussianities,Communications in Theoretical Physics62(2014) 109

    Y. Wang,Inflation, cosmic perturbations and non-gaussianities,Communications in Theoretical Physics62(2014) 109

    work page 2014

  34. [34]

    A. D. Linde,Eternally Existing Selfreproducing Chaotic Inflationary Universe,Phys. Lett. B 175(1986) 395–400

    work page 1986

  35. [35]

    A. S. Goncharov, A. D. Linde and V. F. Mukhanov,The Global Structure of the Inflationary Universe,Int. J. Mod. Phys. A2(1987) 561–591

    work page 1987

  36. [36]

    A. H. Guth,Eternal inflation and its implications,J. Phys. A40(2007) 6811–6826, [hep-th/0702178]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  37. [37]

    The Phase Transition to Eternal Inflation

    P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga,The Phase Transition to Slow-roll Eternal Inflation,JHEP09(2008) 036, [0802.1067]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  38. [38]

    B. J. Carr and S. W. Hawking,Black holes in the early Universe,Mon. Not. Roy. Astron. Soc. 168(1974) 399–415

    work page 1974

  39. [39]

    B. J. Carr,The Primordial black hole mass spectrum,Astrophys. J.201(1975) 1–19

    work page 1975

  40. [40]

    Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -

    M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama,Primordial black holes—perspectives in gravitational wave astronomy,Class. Quant. Grav.35(2018) 063001, [1801.05235]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    A. A. Starobinsky and J. Yokoyama,Equilibrium state of a selfinteracting scalar field in the De Sitter background,Phys. Rev. D50(1994) 6357–6368, [astro-ph/9407016]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  42. [42]

    Miji´ c,Random walk after the big bang,Physical Review D42(1990) 2469

    M. Miji´ c,Random walk after the big bang,Physical Review D42(1990) 2469

    work page 1990

  43. [43]

    A. D. Linde and A. Mezhlumian,Stationary universe,Phys. Lett. B307(1993) 25–33, [gr-qc/9304015]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  44. [44]

    A. D. Linde, D. A. Linde and A. Mezhlumian,From the Big Bang theory to the theory of a stationary universe,Phys. Rev. D49(1994) 1783–1826, [gr-qc/9306035]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  45. [45]

    A. J. Tolley and M. Wyman,Stochastic Inflation Revisited: Non-Slow Roll Statistics and DBI Inflation,JCAP04(2008) 028, [0801.1854]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  46. [46]

    Cohen, D

    T. Cohen, D. Green and A. Premkumar,A tail of eternal inflation,SciPost Phys.14(2023) 109, [2111.09332]

    work page arXiv 2023

  47. [47]

    N. C. Tsamis and R. P. Woodard,Stochastic quantum gravitational inflation,Nuclear Physics B724(2005) 295–328

    work page 2005

  48. [48]

    Gorbenko and L

    V. Gorbenko and L. Senatore,λϕ 4 in dS,1911.00022

    work page arXiv 1911

  49. [49]

    R. P. Woodard,Recent Developments in Stochastic Inflation,2501.15843

    work page arXiv

  50. [50]

    Rey,Dynamics of inflationary phase transition,Nuclear Physics B284(1987) 706–728

    S.-J. Rey,Dynamics of inflationary phase transition,Nuclear Physics B284(1987) 706–728

    work page 1987

  51. [51]

    Nambu and M

    Y. Nambu and M. Sasaki,Stochastic stage of an inflationary universe model,Physics Letters B205(1988) 441–446. – 61 –

    work page 1988

  52. [52]

    Nambu and M

    Y. Nambu and M. Sasaki,Stochastic approach to chaotic inflation and the distribution of universes,Physics Letters B219(1989) 240–246

    work page 1989

  53. [53]

    H. E. Kandrup,Stochastic inflation as a time-dependent random walk,Physical Review D39 (1989) 2245

    work page 1989

  54. [54]

    Habib,Stochastic inflation: Quantum phase-space approach,Physical Review D46(1992) 2408

    S. Habib,Stochastic inflation: Quantum phase-space approach,Physical Review D46(1992) 2408

    work page 1992

  55. [55]

    On the divergences of inflationary superhorizon perturbations

    K. Enqvist, S. Nurmi, D. Podolsky and G. I. Rigopoulos,On the divergences of inflationary superhorizon perturbations,JCAP04(2008) 025, [0802.0395]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  56. [56]

    Infrared Behavior of Quantum Fields in Inflationary Cosmology -- Issues and Approaches: an overview

    B.-L. Hu,Infrared Behavior of Quantum Fields in Inflationary Cosmology – Issues and Approaches: an overview,1812.11851

    work page internal anchor Pith review Pith/arXiv arXiv

  57. [57]

    Baumgart and R

    M. Baumgart and R. Sundrum,De Sitter Diagrammar and the Resummation of Time,JHEP 07(2020) 119, [1912.09502]

    work page arXiv 2020

  58. [58]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi and D. Wands,Quantum diffusion during inflation and primordial black holes,JCAP10(2017) 046, [1707.00537]

    work page arXiv 2017

  59. [59]

    Kuhnel and K

    F. Kuhnel and K. Freese,On Stochastic Effects and Primordial Black-Hole Formation,Eur. Phys. J. C79(2019) 954, [1906.02744]

    work page arXiv 2019

  60. [60]

    J. M. Ezquiaga, J. Garc´ ıa-Bellido and V. Vennin,The exponential tail of inflationary fluctuations: consequences for primordial black holes,JCAP03(2020) 029, [1912.05399]

    work page arXiv 2020

  61. [61]

    Vennin,Stochastic inflation and primordial black holes

    V. Vennin,Stochastic inflation and primordial black holes. PhD thesis, AstroParticule et Cosmologie, France, U. Paris-Saclay, 6, 2020.2009.08715

    work page arXiv 2020

  62. [62]

    Ballesteros, J

    G. Ballesteros, J. Rey, M. Taoso and A. Urbano,Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation,JCAP08(2020) 043, [2006.14597]

    work page arXiv 2020

  63. [63]

    D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg,Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,JCAP05(2022) 027, [2111.07437]

    work page arXiv 2022

  64. [64]

    P. Saha, Y. Tada and Y. Urakawa,Nonlinear Lattice Framework for Inflation: Bridging stochastic inflation and theδNformalism,2604.00978

    work page internal anchor Pith review Pith/arXiv arXiv

  65. [65]

    Nonperturbative stochastic inflation in perturbative dynamical background

    X.-Q. Ye and S.-J. Wang,Nonperturbative stochastic inflation in perturbative dynamical background,2604.15219

    work page internal anchor Pith review Pith/arXiv arXiv

  66. [66]

    Noise and Fluctuations in Semiclassical Gravity

    E. Calzetta and B. L. Hu,Noise and fluctuations in semiclassical gravity,Phys. Rev. D49 (1994) 6636–6655, [gr-qc/9312036]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  67. [67]

    Lagrangian Formulation of Stochastic Inflation: A Recursive Approach

    L. Perreault Levasseur,Lagrangian formulation of stochastic inflation: Langevin equations, one-loop corrections and a proposed recursive approach,Phys. Rev. D88(2013) 083537, [1304.6408]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  68. [68]

    A new algorithm for calculating the curvature perturbations in stochastic inflation

    T. Fujita, M. Kawasaki, Y. Tada and T. Takesako,A new algorithm for calculating the curvature perturbations in stochastic inflation,JCAP12(2013) 036, [1308.4754]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  69. [69]

    Non-perturbative approach for curvature perturbations in stochastic-$\delta N$ formalism

    T. Fujita, M. Kawasaki and Y. Tada,Non-perturbative approach for curvature perturbations in stochasticδNformalism,JCAP10(2014) 030, [1405.2187]. – 62 –

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  70. [70]

    Backreaction and Stochastic Effects in Single Field Inflation

    L. Perreault Levasseur and E. McDonough,Backreaction and Stochastic Effects in Single Field Inflation,Phys. Rev. D91(2015) 063513, [1409.7399]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  71. [71]

    Correlation Functions in Stochastic Inflation

    V. Vennin and A. A. Starobinsky,Correlation Functions in Stochastic Inflation,Eur. Phys. J. C75(2015) 413, [1506.04732]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  72. [72]

    Grain and V

    J. Grain and V. Vennin,Stochastic inflation in phase space: is slow roll a stochastic attractor?,JCAP05(2017) 045, [1703.00447]

    work page arXiv 2017

  73. [73]

    Stochastic Ultra Slow Roll Inflation

    H. Firouzjahi, A. Nassiri-Rad and M. Noorbala,Stochastic Ultra Slow Roll Inflation,JCAP 01(2019) 040, [1811.02175]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  74. [74]

    Inflationary stochastic anomalies

    L. Pinol, S. Renaux-Petel and Y. Tada,Inflationary stochastic anomalies,Class. Quant. Grav. 36(2019) 07LT01, [1806.10126]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  75. [75]

    Firouzjahi, A

    H. Firouzjahi, A. Nassiri-Rad and M. Noorbala,Stochastic nonattractor inflation,Phys. Rev. D102(2020) 123504, [2009.04680]

    work page arXiv 2020

  76. [76]

    Pattison, V

    C. Pattison, V. Vennin, D. Wands and H. Assadullahi,Ultra-slow-roll inflation with quantum diffusion,JCAP04(2021) 080, [2101.05741]

    work page arXiv 2021

  77. [77]

    Cruces and C

    D. Cruces and C. Germani,Stochastic inflation at all order in slow-roll parameters: Foundations,Phys. Rev. D105(2022) 023533, [2107.12735]

    work page arXiv 2022

  78. [78]

    Cohen, D

    T. Cohen, D. Green, A. Premkumar and A. Ridgway,Stochastic Inflation at NNLO,JHEP09 (2021) 159, [2106.09728]

    work page arXiv 2021

  79. [79]

    Towards Stochastic Inflation in Higher-Curvature Gravity

    Y. Aldabergenov, D. Ding, W. Lin and Y. Wan,Towards Stochastic Inflation in Higher-Curvature Gravity,2506.21423

    work page internal anchor Pith review Pith/arXiv arXiv

  80. [80]

    R. K. Panda, S. Panda and A. Tinwala,Revisiting Lagrangian Formulation of Stochastic inflation,2510.03171

    work page arXiv

Showing first 80 references.