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A simple construction of the sine-Gordon model via stochastic quantization

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Pith's one-line read The paper proves a simple PDE construction of the sine-Gordon measure for $0<\beta^2<4\pi$, in finite and infinite volume, via stochastic quantization.

desk verdict A solid stochastic-quantization compactness result with a real presentation gap: the abstract oversells subsequential tightness as construction of the sine-Gordon measure. read the letter →

arxiv 2412.16404 v1 pith:N7JADR74 submitted 2024-12-20 math.PR math-phmath.APmath.MP

Massimiliano Gubinelli , Martin Hairer , Tadahiro Oh , Younes Zine This is my paper · ORCID
classification math.PRmath-phmath.APmath.MP MSC 81T0860H1535K0535L71
keywords sine-GordonmodelstochasticquantizationimaginaryGaussianmultiplicativechaostightnessparabolichyperbolicrenormalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that for $0<\beta^2<4\pi$, the renormalized truncated sine-Gordon measures $\rho_{L,N}$ on a two-dimensional torus are tight, so they admit weak limit points; the same compactness argument, applied along growing tori, yields a limiting probability measure on $\mathbb{R}^2$. The proof works through the parabolic sine-Gordon equation, a stochastic heat flow for which the sine-Gordon measure is the formal invariant measure, and controls the flow by a single uniform bound on a renormalized exponential of the stochastic convolution, the imaginary Gaussian multiplicative chaos. The same deterministic estimate, run through the damped wave propagator, gives pathwise global well-posedness of the hyperbolic sine-Gordon model for $0<\beta^2<2\pi$. The appeal of the construction is that it uses only standard heat-flow smoothing, product estimates for rough distributions, and an invariance argument, rather than a dedicated measure-theoretic construction.

What carries the argument

The load-bearing object is the renormalized imaginary Gaussian multiplicative chaos $\Theta_{L,N}=e^{\frac{\beta^2}{2}\sigma_{L,N}}e^{i\beta\Psi_{L,N}}$, where $\Psi$ is the stationary solution of the linear heat equation with space-time white noise and $\sigma_{L,N}=\mathbb{E}[(\Pi_N\Psi)^2]\approx(2\pi)^{-1}\log N$ is the renormalization counterterm. Proposition 2.5 supplies the crucial quantitative estimate: for $\beta^2<4\pi$, $\Theta_{L,N}$ is Cauchy in $L^p(C([0,T];C^{-\alpha}))$ and its $C^{-\alpha}$ norm localized by a disk $B$ has $p$-th moment bounded by $C\min(|B|,L^2)$, uniformly in $N$ and $L$. This estimate feeds a deterministic global-in-time a priori bound (Proposition 3.1) for the equation $\partial_t v+(1-\Delta)v=\sum_{\kappa\in\{+, -\}}e^{i\kappa\beta v}\Theta_\kappa$, proved with heat-kernel smoothing, a product estimate for distributions of opposite regularity, and the boundedness of $e^{\pm i\beta v}$. Combining the a priori bound with invariance of the truncated measure under the truncated flow (Lemma 1.3) gives uniform moment bounds on the initial data and hence tightness.

What would settle it

Compute the $p$-th moment of the $C^{-\alpha}$ norm of $\chi_B\Theta_{L,N}$ on a large torus with $L\gg 1$, at $\beta^2$ close to $4\pi$, and compare it with $C\min(|B|,L^2)$: if the moment grows faster than $L^2$ or fails to be uniform in $N$, Proposition 2.5 is false and the proof of Theorems 1.1 and 1.2 does not go through. A direct check of the sketched final paragraph of Section 2.3 would also settle whether the proof of this estimate is complete.

Watch

Extended reading notes

Core claim

The central claim is that below the first threshold, $\beta^2<4\pi$, the sequence of renormalized truncated sine-Gordon measures $\rho_{L,N}$ on a fixed two-dimensional torus is tight in $C^{-\delta}$, so it has weakly convergent subsequences; and along a subsequence of growing tori these limits converge weakly on $\mathbb{R}^2$ to a probability measure $\rho^\beta$. The proof transfers the measure renormalization $\gamma_{L,N}=e^{\frac{\beta^2}{2}\sigma_{L,N}}$ to the dynamics, writes the truncated solution as $u=\Psi+v$ with $\Psi$ the stationary stochastic convolution of the linear heat equation, and reduces everything to a deterministic global a priori bound for a nonlinear heat equation forced by products $e^{\pm i\beta v}\Theta_{L,N}$. The only stochastic input is a uniform $L^p$ bound, uniform in the truncation $N$ and the torus size $L$, for the imaginary Gaussian multiplicative chaos $\Theta_{L,N}=e^{\frac{\beta^2}{2}\sigma_{L,N}}e^{i\beta\Psi_{L,N}}$ and for its spatial localizations. The same a priori bound, adapted to the damped wave propagator, yields pathwise global well-posedness of the hyperbolic sine-Gordon model for $\beta^2<2\pi$.

Load-bearing premise

The load-bearing premise is that the renormalized exponential $\Theta_{L,N}$ of the stochastic convolution has $p$-th moments bounded by the volume of the spatial region, uniformly in the frequency cutoff $N$ and the torus size $L$; if that bound grows faster than $\min(|B|,L^2)$, the tightness argument in Theorems 1.1 and 1.2 collapses.

Editorial extensions

If this is right

  • For every $0<\beta^2<4\pi$ and every fixed torus size $L$, the family $\{\rho_{L,N}\}_N$ is tight in $C^{-\delta}$, so a subsequence converges weakly to a probability measure $\rho_L^\beta$ on $C^{-\delta}(\mathbb{T}_L^2)$.
  • Along a suitable subsequence of growing tori, the measures $\rho_{L_j}^\beta$ converge weakly on $\mathcal{D}'(\mathbb{R}^2)$ to a probability measure $\rho^\beta$, giving an infinite-volume sine-Gordon measure below the first threshold.
  • The parabolic sine-Gordon model is pathwise globally well-posed for $\beta^2<4\pi$: the truncated dynamics converge in probability to a limiting solution $u=\Psi+v$.
  • The damped hyperbolic sine-Gordon model is pathwise globally well-posed for $0<\beta^2<2\pi$ with deterministic initial data in $H^s\times H^{s-1}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the deterministic bound only uses the boundedness and Lipschitz character of $e^{\pm i\beta v}$, the same scheme should produce invariant measures for stochastic heat equations with other bounded Lipschitz interactions, with the admissible range set by the regularity of the corresponding multiplicative chaos.
  • The uniform volume growth $\min(|B|,L^2)$ in Proposition 2.5 suggests the method is not special to sine-Gordon: exponential (Liouville-type) interactions on large tori should be treatable by the same localization argument.
  • The paper leaves open whether the subsequential limits coincide with the sine-Gordon measure constructed by other methods; showing convergence of characteristic functionals or matching Dirichlet forms would close the gap.
  • For the hyperbolic model, the restriction $\beta^2<2\pi$ comes from the single degree of smoothing of the damped wave propagator; reaching $4\pi$ would likely require negative temporal regularity or additional stochastic objects.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a stochastic-quantization construction of the Euclidean sine-Gordon measure for 0 < beta^2 < 4*pi, in finite volume and, by a large-torus limit, on R^2. The main results are Theorem 1.1 (subsequential weak convergence of the renormalized truncated measures rho_{L,N} to some probability measure rho_L on C^{-delta}(T^2_L)), Theorem 1.2 (an analogous subsequential limit of the torus-periodic measures in a weighted Besov space on R^2), and Theorem A.1 (pathwise global well-posedness of the damped hyperbolic sine-Gordon model for 0 < beta^2 < 2*pi). The proofs combine a deterministic global a priori bound for a nonlinear heat equation with distributional coefficients (Proposition 3.1), regularity estimates for the truncated stochastic convolution and the imaginary Gaussian multiplicative chaos (Propositions 2.4 and 2.5), and a weighted-space compactness argument in Section 4.

Significance. The deterministic a priori bound in Proposition 3.1 is clean and appears correct; the idea of exploiting the boundedness and Lipschitz property of the sine nonlinearity is a genuine simplification for the parabolic equation. The hyperbolic result in Appendix A is also a useful contribution. However, the paper's advertised claim of 'construction of the sine-Gordon measure' is stronger than what Theorems 1.1 and 1.2 establish: they provide only subsequential compactness, with no identification of any limit point as the sine-Gordon measure. In addition, the crucial L-dependence of the chaos bounds in Proposition 2.5 is only sketched. For these reasons the significance is conditional: if the missing identifications and estimates are supplied or the claims are revised to 'tightness' and 'subsequential limits', the paper would provide a valuable PDE route to compactness and well-posedness results, but in its current form the central interpretation overreaches the proofs.

major comments (2)
  1. [Section 1.1, Theorems 1.1-1.2, and Section 4] Theorems 1.1 and 1.2 establish only subsequential weak convergence: Theorem 1.1 returns a subsequence {rho_{L,N_j}} converging to some probability measure rho_L, and Theorem 1.2 returns a further subsequence of torus sizes, again without uniqueness or identification. The text nevertheless refers to 'the sine-Gordon measure rho_L^beta constructed in Theorem 1.1' immediately after Theorem 1.1, and the proof of Theorem 1.2 near (4.22) uses the same phrase. No argument is supplied that identifies any subsequential limit with the known sine-Gordon measure: there is no proof of convergence of the full sequence, no analysis of correlation functions or characteristic functionals, and no uniqueness result for invariant measures of the limiting SPDE. Consequently, the abstract's phrase 'construction of the sine-Gordon measure' and the title's 'construction' overstate what is proven. The compactness statements themselves may be valuable, but the paper should either prove the identification or revise the claims and interpretation accordingly.
  2. [Section 2.3, Proposition 2.5] The proof of the uniform L-dependence in (2.18)-(2.19) is only sketched. The final paragraph of Section 2.3 reduces the problem to the local estimate (2.40) via a covering argument and then says 'The rest of the proof follows from similar arguments'; the actual covering argument converting the O(1) local bounds into the L^2 factor in (2.18), and the localization argument producing min(|B|,L^2) in (2.19), are not written out. These estimates are the only place where the L-dependence used in (4.19)-(4.20) for Theorem 1.2 is derived, so if the covering argument required an extra power of L or a logarithmic factor, the uniform bound (4.22) and hence Theorem 1.2 would fail. Please provide a complete proof of Proposition 2.5, or state the L-dependence explicitly as an assumption with a precise reference. In particular, the comparison in (2.42) invoking (2.18) for a fixed small L0 should be made fully explicit so that the argument is not circular.
minor comments (5)
  1. [Eq. (2.42)] In the last display of Section 2.3, the factor 'e^{-i Psi^2_{L0,N}}' appears without the parameter beta; it should read 'e^{-i beta Psi^2_{L0,N}}' to match the preceding definitions.
  2. [Lemma 2.6] The proof of Lemma 2.6 says 'By Plancherel's identity and Young's inequality' for what appears to be an L^p estimate with general p. Plancherel is specific to p=2; the estimate (2.25) follows from Young's convolution inequality and the Fourier multiplier bound, and the wording should be corrected.
  3. [Proof of Theorem 1.1 and Eq. (1.13)] Proposition 3.1 is stated for (3.2) without the projector Pi_N, but (1.13) contains Pi_N in front of the Duhamel term. The application is justified if Pi_N is uniformly bounded on the relevant Besov spaces, but this should be stated explicitly.
  4. [Lemma 4.1] The compact embedding lemma for the weighted Besov spaces is stated without proof ('the required modification is straightforward'). Since this embedding is used in the tightness argument for Theorem 1.2, a proof or a precise reference would be appropriate.
  5. [Section 3.2] Tightness on C^{-delta}(T_L^2) is deduced from the moment bound sup_N E ||u_{L,N}(0)||^p < infinity. Because C^{-delta} is not a separable space and balls are not compact, the argument relies on the fact that the same bound holds for a slightly smaller delta and on the compact embedding C^{-delta'} -> C^{-delta}; the authors should spell this out rather than say only that the choice of small delta was arbitrary.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the truncated measures and invariant dynamics yield the compactness bounds directly; the main weaknesses (unproved identification with the sine-Gordon measure and sketched chaos bounds) are completeness/correctness gaps, not circular reductions.

full rationale

The central derivation is not circular. The target formal density in (1.1) is fixed before the construction, and the truncated approximating measures are defined independently in (1.5) from the massive Gaussian free field and the explicit renormalization γ_L,N = exp((β^2/2)σ_L,N), with σ computed in (1.4). Lemma 1.3 verifies invariance of ρ_L,N under the truncated dynamics (1.8) by a direct computation; it does not assume the limiting measure. The stochastic inputs Ψ and Θ in Propositions 2.4 and 2.5 are obtained from Gaussian estimates (Wick renormalization, heat-kernel bounds, and localization/covering arguments). For the base L ~ 1 estimates the paper cites [31,36], which are prior theorems on the dynamical sine-Gordon model and imaginary Gaussian multiplicative chaos; although the authorship overlaps, those theorems are not the compactness claim and are not fitted to it. The deterministic a priori bound in Proposition 3.1 and the bootstrap in (3.10)-(3.11) use only semigroup estimates, the boundedness of e^{±iβv}, and invariance; the conclusion is a tightness statement via Prokhorov's theorem, not a measure whose definition presupposes the theorem. What the paper does not do is identify the subsequential limits with the known sine-Gordon measure: Section 1.1 writes 'let ρ_L^β be the sine-Gordon measure ... constructed in Theorem 1.1' even though Theorem 1.1 supplies only a subsequential weak limit, and Section 4 repeats this ('Now, let ρ_L^β be the weak limit ...'). This is an overstatement and an unproved identification, but not a circular equation. Likewise, Proposition 2.5's large-torus extension is only sketched ('The rest of the proof follows from similar arguments'), a completeness gap, and Appendix A explicitly disclaims a measure construction via the hyperbolic model (Remark A.5). These are correctness risks, not steps that reduce by construction to their own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants; model parameters beta and gamma are inputs, and the renormalization constant gamma_{L,N}=exp(beta^2 sigma_{L,N}/2) is fixed by the Wick prescription (1.3)-(1.4). The proof uses standard Besov/paraproduct calculus, the stationary stochastic convolution, and the imported invariance Lemma 1.3. No new physical entities are introduced.

assumptions (4)
  • standard math Littlewood-Paley/Besov calculus on tori and R2, including Schauder estimates, Bony product estimates, and embeddings (Lemmas 2.1-2.3, 2.6)
    Invoked throughout Sections 2 and 3 to control products of distributions and heat semigroup smoothing.
  • domain assumption Existence of a unique stationary solution Psi_L to the linear equation (d_t+1-Delta)Psi = sqrt(2) xi on R+ x T2_L
    Used in the Da Prato-Debussche decomposition (1.10); its regularity is quantified in Proposition 2.4.
  • domain assumption Standard Wick renormalization: gamma_{L,N}=exp(beta^2 sigma_{L,N}/2) with sigma_{L,N} ~ (1/2pi) log N
    Defined in (1.3)-(1.4); the convergence of Theta depends on this exact choice, which is standard and not fitted.
  • domain assumption Lemma 1.3: the truncated measure rho_{L,N} is invariant under the truncated dynamics (1.8)
    Stated without proof, citing [37, Subsection 5.2]; the tightness argument (3.11) uses it to replace the law at time T with the initial law.

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Pith. "Pith review of A simple construction of the sine-Gordon model via stochastic quantization." pith.science (2026). https://pith.science/paper/N7JADR74

@misc{pith2026241216404,
  author       = {Pith},
  title        = {Pith review of: A simple construction of the sine-Gordon model via stochastic quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7JADR74}},
  note         = {Machine review of arXiv:2412.16404}
}
abstract

We present a simple PDE construction of the sine-Gordon measure below the first threshold ($\be^2 < 4\pi$), in both the finite and infinite volume settings, by studying the corresponding parabolic sine-Gordon model. We also establish pathwise global well-posedness of the hyperbolic sine-Gordon model in finite volume for $\be^2 < 2\pi$.

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Reference graph

Works this paper leans on

43 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Albeverio, S

    S. Albeverio, S. Kusuoka, The invariant measure and the flow associated to the Φ 4 3-quantum field model , Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020), no. 4, 1359–142 7

    work page 2020

  2. [2]

    Albeverio, S

    S. Albeverio, S. Kusuoka, Construction of a non-Gaussian and rotation-invariant Φ 4-measure and asso- ciated flow on R3 through stochastic quantization , arXiv:2102.08040 [math.PR]

    arXiv

  3. [3]

    Aronszajn, K

    N. Aronszajn, K. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475

    work page 1961

  4. [4]

    Barashkov, A stochastic control approach to Sine Gordon EQFT , arXiv:2203.06626 [math.PR]

    N. Barashkov, A stochastic control approach to Sine Gordon EQFT , arXiv:2203.06626 [math.PR]

    arXiv

  5. [5]

    Barashkov, M

    N. Barashkov, M. Gubinelli, A variational method for Φ 4 3, Duke Math. J. 169 (2020), no. 17, 3339–3415

    2020

  6. [6]

    Bauerschmidt, T

    R. Bauerschmidt, T. Bodineau, Log-Sobolev inequality for the continuum sine-Gordon mode l, Comm. Pure Appl. Math. 74 (2021), no. 10, 2064–2113

    work page 2021

  7. [7]

    B´ enyi, T

    ´A. B´ enyi, T. Oh, T. Zhao, Fractional Leibniz rule on the torus, Proc. Amer. Math. Soc. (2024). https://doi.org/10.1090/proc/17007

    doi:10.1090/proc/17007 2024

  8. [8]

    Bony, Calcul symbolique et propagation des singularit´ es pour le s ´ equations aux d´ eriv´ ees partielles non lin´ eaires,Ann

    J.-M. Bony, Calcul symbolique et propagation des singularit´ es pour le s ´ equations aux d´ eriv´ ees partielles non lin´ eaires,Ann. Sci. ´Ecole Norm. Sup. 14 (1981), no. 2, 209–246

    work page 1981

Show all 43 references
  1. [9]

    Bourgain, Periodic nonlinear Schr¨ odinger equation and invariant me asures, Comm

    J. Bourgain, Periodic nonlinear Schr¨ odinger equation and invariant me asures, Comm. Math. Phys. 166 (1994), no. 1, 1–26

    1994

  2. [10]

    Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr¨ o dinger equation , Comm

    J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr¨ o dinger equation , Comm. Math. Phys. 176 (1996), no. 2, 421–445

    1996

  3. [11]

    Bringmann, S

    B. Bringmann, S. Cao, Global well-posedness of the dynamical sine-Gordon model u p to 6π , arXiv:2410.15493 [math.AP]

    arXiv

  4. [12]

    Chandra, G.L

    A. Chandra, G.L. Feltes, H. Weber, A priori bounds for 2-d generalised parabolic Anderson mode l, arXiv:2402.05544 [math.AP]

  5. [13]

    Chandra, M

    A. Chandra, M. Hairer, H. Shen, The dynamical sine-Gordon model in the full subcritical reg ime, arXiv:1808.02594 [math.PR]

    arXiv

  6. [14]

    Christ, M

    M. Christ, M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal. 100 (1991), 87–109

    1991

  7. [15]

    Da Prato, A

    G. Da Prato, A. Debussche, Strong solutions to the stochastic quantization equations , Ann. Probab. 31 (2003), no. 4, 1900–1916

    2003

  8. [16]

    Dimock, T.R

    J. Dimock, T.R. Hurd, A renormalization group analysis of the Kosterlitz-Thoule ss phase , Comm. Math. Phys. 137 (1991), no. 2, 263–287

    1991

  9. [17]

    Dimock, T.R

    J. Dimock, T.R. Hurd, Construction of the two-dimensional sine-Gordon model for β ă 8π , Comm. Math. Phys. 156 (1993), no. 3, 547–580

    1993

  10. [18]

    Dimock, T.R

    J. Dimock, T.R. Hurd, Sine-Gordon revisited, Ann. Henri Poincar´ e 1 (2000), no. 3, 499–541

    2000

  11. [19]

    sine-Gordon

    J. Fr¨ ohlich,Quantized “sine-Gordon ” equation with a nonvanishing mass term in two space-time dimen- sions, Phys. Rev. Lett. 34 (1975), 833–836. SINE-GORDON MODEL VIA STOCHASTIC QUANTIZATION 25

    1975

  12. [20]

    Fr¨ ohlich, Y.M

    J. Fr¨ ohlich, Y.M. Park, Remarks on exponential interactions and the quantum sine-G ordon equation in two space-time dimensions , Helv. Phys. Acta 50 (1977), no. 3, 315–329

    1977

  13. [21]

    Gatto, Product rule and chain rule estimates for fractional deriva tives on spaces that satisfy the doubling condition , J

    A.E. Gatto, Product rule and chain rule estimates for fractional deriva tives on spaces that satisfy the doubling condition , J. Funct. Anal. 188 (2002), no. 1, 27–37

    2002

  14. [22]

    Glimm, A

    J. Glimm, A. Jaffe, Quantum physics. A functional integral point of view , Second edition. Springer-Verlag, New York, 1987. xxii+535 pp

    1987

  15. [23]

    Grafakos, Classical Fourier analysis , Third edition

    L. Grafakos, Classical Fourier analysis , Third edition. Graduate Texts in Mathematics, 249. Spring er, New York, 2014. xviii+638 pp

    2014

  16. [24]

    Grafakos, Modern Fourier analysis, Third edition

    L. Grafakos, Modern Fourier analysis, Third edition. Graduate Texts in Mathematics, 250. Spring er, New York, 2014. xvi+624 pp

    2014

  17. [25]

    Gubinelli, P

    M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75 pp

    2015

  18. [26]

    Gubinelli, M

    M. Gubinelli, M. Hofmanov´ a, A PDE construction of the Euclidean φ 4 3 quantum field theory , Comm. Math. Phys. 384 (2021), no. 1, 1–75

    2021

  19. [27]

    Gubinelli, H

    M. Gubinelli, H. Koch, T. Oh, Renormalization of the two-dimensional stochastic nonlin ear wave equa- tions, Trans. Amer. Math. Soc. 370 (2018), no 10, 7335–7359

    2018

  20. [28]

    Gubinelli, S.-J

    M. Gubinelli, S.-J. Meyer, The FBSDE approach to sine-Gordon up to 6π , arXiv:2401.13648 [math-ph]

    arXiv

  21. [29]

    Hairer, A theory of regularity structures, Invent

    M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504

    2014

  22. [30]

    Hairer, M

    M. Hairer, M. D. Ryser, H. Weber, Triviality of the 2D stochastic Allen-Cahn equation , Electron. J. Probab. 17 (2012), no. 39, 14 pp

    2012

  23. [31]

    Hairer, H

    M. Hairer, H. Shen, The dynamical sine-Gordon model , Comm. Math. Phys. 341 (2016), no. 3, 933–989

    2016

  24. [32]

    McKean, Statistical mechanics of nonlinear wave equations

    H.P. McKean, Statistical mechanics of nonlinear wave equations. IV. Cub ic Schr¨ odinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491. Erratum: Statistical mechanics of nonlinear wave equation s. IV. Cubic Schrodinger, Comm. Math. Phys. 173 (1995), no. 3, 675

    1995

  25. [33]

    Mourrat, H

    J.-C. Mourrat, H. Weber, Global well-posedness of the dynamic Φ 4 model in the plane , Ann. Probab.45 (2017), no. 4, 2398–2476

    2017

  26. [34]

    Mourrat, H

    J.-C. Mourrat, H. Weber, The dynamic Φ 4 3 model comes down from infinity , Comm. Math. Phys. 356 (2017), no. 3, 673–753

    2017

  27. [35]

    Parisi, Y.S

    G. Parisi, Y.S. Wu, Perturbation theory without gauge fi xing, Sci. Sinica 24 (1981), no. 4, 483–496

    1981

  28. [36]

    T. Oh, T. Robert, P. Sosoe, Y. Wang, Invariant Gibbs dynamics for the dynamical sine-Gordon mod el, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1450-146 6

    2021

  29. [37]

    T. Oh, T. Robert, Y. Wang, On the parabolic and hyperbolic Liouville equations , Comm. Math. Phys. 387 (2021), no. 3 1281–1351

    2021

  30. [38]

    T. Oh, L. Tolomeo, Y. Wang, G. Zheng, Hyperbolic P pΦ q2-model on the plane , arXiv:2211.03735 [math.AP]

  31. [39]

    T. Oh, Y. Zine, Hyperbolic sine-Gordon model beyond the first threshold , preprint

  32. [40]

    Ryang, T

    S. Ryang, T. Saito, K. Shigemoto, Canonical stochastic quantization , Progr. Theoret. Phys. 73 (1985), no. 5, 1295–1298

    1985

  33. [41]

    H. Shen, R. Zhu, X. Zhu Global well-posedness for 2D generalized parabolic Anders on model via paracon- trolled calculus, arXiv:2402.19137 [math.AP]

    arXiv

  34. [42]

    Staffilani, The initial value problem for some dispersive differential eq uations, Thesis (Ph.D.) – The University of Chicago

    G. Staffilani, The initial value problem for some dispersive differential eq uations, Thesis (Ph.D.) – The University of Chicago. 1995. 88 pp

    1995

  35. [43]

    Zine, Smoluchowski-Kramers approximation for singular stochas tic wave equations in two dimensions , arXiv:2206.08717 [math.AP]

    Y. Zine, Smoluchowski-Kramers approximation for singular stochas tic wave equations in two dimensions , arXiv:2206.08717 [math.AP]. Massimiliano Gubinelli, Mathematical Institute, Univers ity of Oxford, Andrew Wiles Build- ing, Radcliffe Observatory Quarter, Woodstock Road, Ox...

    arXiv

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