Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

Shyam Popat

REVIEW 4 major objections 4 minor 66 references

Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

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Pith's one-line read This paper proves a central limit theorem and a large-deviations rate function for generalized Dean–Kawasaki SPDEs on bounded domains with Dirichlet boundary data.

desk verdict Genuine bounded-domain extension of the torus CLT/LDP results for Dean-Kawasaki SPDEs; the singular CLT rests on a sketched Moser-iteration estimate (Theorem 6) that needs a completed proof before the paper is fully verified. read the letter →

arxiv 2504.17094 v1 pith:PZTW4WKP submitted 2025-04-23 math.PR math.AP

Shyam Popat This is my paper · ORCID

classification math.PRmath.AP MSC 60F0560F1060H1582B2182B3160K3535Q70

keywords CentrallimittheoremLargedeviationsprincipleDean–KawasakiequationDirichletboundaryconditionsStochastickineticsolutionsInteractingparticlesystemsZerorangeprocess

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

Generalized Dean–Kawasaki SPDEs describe how the density of an interacting particle system fluctuates when driven by small multiplicative noise. The paper proves two fluctuation results for these equations on arbitrary $C^2$ bounded domains with Dirichlet boundary conditions: under the joint scaling $\varepsilon K^{d+2}\to 0$, the rescaled fluctuation $v^{\varepsilon,K}=\varepsilon^{-1/2}(\rho^{\varepsilon,K}-\bar\rho)$ converges in probability to the solution of the linearized SPDE (a central limit theorem), and the solutions satisfy a large deviations principle with the same variational rate function that governs the zero-range process. This matters because the boundary condition is what makes a particle model describe absorption or injection of particles at the boundary, turning fluctuation results on the torus into results for realistic bounded regions. The technical price is a set of new $L^p$ energy estimates and an $L^\infty$ positivity estimate that keep the singular square-root diffusion away from its zero set.

What carries the argument

The load-bearing mechanism is the theory of stochastic kinetic solutions: the equation is renormalized away from the zero set of the density, where the square-root diffusion coefficient is singular, and the Dirichlet condition is encoded through $\Phi(\rho^{\varepsilon,K})|_{\partial U}=\bar f$. On top of that solution theory, the CLT rests on three devices: Assumption 2, which fixes the initial and boundary data to the same positive constant $M$ and makes the hydrodynamic limit the constant $M$; the auxiliary potentials $\Theta_{\Phi,p}$, defined by $\Theta_{\Phi,p}'(\xi)=(\xi-\bar\rho)^{(p-2)/2}\Phi'(\xi)^{1/2}$, which convert the diffusion into gradients whose $L^2$ norm is controlled by the energy estimates; and a Moser-iteration $L^\infty$ estimate showing the solution stays above $M/2$ with high probability, so the singular coefficient can be replaced by a bounded one. For the large deviations principle, the weak approach of [BDM08] represents the noise through a controlled SPDE; the skeleton limit of the controlled equation is a parabolic-hyperbolic PDE whose solution map is compact in $L^1(U\times[0,T])$, and the rate function is the minimal $L^2$ cost of a control driving that skeleton equation.

What would settle it

A direct numerical experiment would settle the scaling claim: take the Dirichlet-eigenfunction noise of Example 1 and keep $\varepsilon K^{d+2}=\delta>0$ fixed as $\varepsilon\to0$; Theorem 7's error bound has prefactors $\|F_3^K\|_{L^\infty}\sim K^{d+2}$, so the right-hand side should not vanish and $v^{\varepsilon,K}$ should fail to converge to the linearized SPDE if the estimates are sharp. A complementary analytic check is homogeneous Dirichlet data $\Phi(\rho)|_{\partial U}=0$ with fast diffusion $\Phi(\xi)=\xi^m$, $m<1$, where deterministic finite-time extinction is known; then Theorem 6's positivity estimate is false, so Theorem 7 cannot cover that boundary data.

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Extended reading notes

Core claim

The paper's central claim is twofold. Theorem 7 states that for stochastic kinetic solutions of (1) under Assumptions 1, 2, 3 and 6, the process $v^{\varepsilon,K}=\varepsilon^{-1/2}(\rho^{\varepsilon,K}-\bar\rho)$ converges in probability in $L^2([0,T];H^{-s}(U))$ for every $s>d/2$ to the unique strong solution of the linearized SPDE $\partial_t v=\Delta(\Phi'(\bar\rho)v)-\nabla\cdot(\sigma(\bar\rho)\dot\xi+\nu'(\bar\rho)v)$ with zero initial and boundary data, whenever the eigenfunction noise is truncated so that $\varepsilon K^{d+2}\to 0$. Theorem 8 states that for boundary data $\bar f\in H^1(\partial U)$ and initial data with finite $\Phi$-entropy, the solutions satisfy a large deviations principle on $L^1(U\times[0,T])$ with rate function $$I_{\bar f,\rho_0}(\rho)=\frac12\inf_{g\in $L^{2}$(U\times[0,T];\mathbb R^d)}\left\{\|g\|^2_{$L^{2}$(U\times[0,T];\mathbb R^d)} : \partial_t\rho=\$\Delta$\Phi(\rho)-\nabla\cdot(\$\sigma$(\rho)g+\nu(\rho)),\ \Phi(\rho)|_{\partial U}=\bar f,\ \rho(\cdot,0)=\rho_0\right\}.$$ This rate function is exactly the one established for the zero-range process in earlier work, so the SPDE reproduces the particle system both in first-order fluctuations and in the exponential cost of rare events.

Load-bearing premise

The entire CLT proof depends on Assumption 2, that the initial density and the Dirichlet boundary value are one and the same random positive constant $M$; if the boundary value differs from the initial data, the hydrodynamic limit is no longer constant, and the paper says only $L^2$ estimates are available, so the $L^4$ and $L^{2\beta+4}$ bounds at the heart of the CLT cannot be obtained.

Editorial extensions

If this is right

On any $C^2$ bounded domain, typical fluctuations of the density are asymptotically Gaussian and governed by the linearized SPDE (4), so SPDE simulations can replace expensive particle simulations in the central-limit regime.
Because the rate function matches the zero-range-process rate, probabilities of rare density profiles can be computed from the variational problem $I_{\bar f,\rho_0}$ instead of by simulating huge particle systems.
Dirichlet boundaries correspond to absorption or injection of particles, so the results cover heat reservoirs, traffic inflow and outflow, wealth entering or leaving a system, and population immigration or emigration on bounded regions.
The joint scaling $\varepsilon K^{d+2}\to 0$ gives a concrete prescription: for noise amplitude $\varepsilon$, the number $K$ of eigenmodes must grow faster than $\varepsilon^{-1/(d+2)}$ for the limit theorems to hold.
The large deviations principle is uniform over compact sets of initial data, while the paper proves a structural obstruction, boundary normal-derivative terms, preventing uniformity over boundary data.

Reading between the lines

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

One can test sharpness of the scaling: if $\varepsilon K^{d+2}\to c>0$, the error prefactors in Theorem 7 should fail to vanish, so $v^{\varepsilon,K}$ should stop converging; this would show the scaling is necessary rather than merely sufficient.
The CLT is restricted to random constant initial and boundary data, and the paper's Remark 15 says only $L^2$ bounds are available without this; a non-constant boundary profile with a nontrivial stationary state likely needs a genuinely different fluctuation theory with boundary layers.
The successful rate-function match with zero-range processes suggests the same variational rate function might describe other boundary-driven particle models such as exclusion processes with reservoirs, but establishing that would require a separate particle-side large-deviations argument not attempted here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main thing you should know: this is a real extension, not a repackaging. The torus CLT (Clini-Fehrman) and torus LDP (Fehrman-Gess) are moved to C^2 bounded domains with Dirichlet boundary data, boundary terms included, full range m in (0,infinity) for Phi(xi)=xi^m, and the rate function matches the zero-range process on the bounded domain. Theorems 7 and 8 are new statements, and the LDP half is largely written out.

Credit where due: the energy estimates (Propositions 4 and 5) are detailed, the joint scaling in Remark 17 is explicit (epsilon K^{d+2} -> 0 for the eigenfunction noise), and the paper is upfront about Assumption 2 -- the CLT only treats constant initial/boundary data M, and Remark 15 admits the L^4 and L^{2beta+4} bounds in the CLT proof would fail otherwise. The abstract's Dirichlet-generality is really delivered by the LDP section, not the CLT, which is a mismatch worth knowing. No circularity: nothing is fitted, and the reliance on prior work (Pop25, DFG20, FG23) is legitimate.

Now the soft spot. The stress-test note lands. Theorem 6 is the load-bearing step for the singular CLT, and the paper itself says the 'improved estimate' (52) follows 'as in Theorem 3.9 of [DFG20]' -- that is a sketch. The Moser iteration starts at p_0 = 0, which means applying (52) with p = 2 leaves a factor ||(rho-M)_-||_{L^{p-2}} with p-2 = -2, an undefined 'norm'; the base case is formal. The paper does bound the constant product in (53), so I would guess the result is true and the gap is fixable, but as written the CLT in probability rests on an unfinished proof, and the bound E||(rho^{epsilon,K}-M)_-||_{L^infinity} <= c(epsilon(...))^gamma is not established. That is the main thing a referee should demand.

Minor: Proposition 6's uniqueness proof has a duplicated summation index (the double sum over j and j,k) -- cosmetic. Theorem 9 delegates skeleton well-posedness to 'adapting' FG23/Pop25; acceptable, but thin.

Bottom line: for Dean-Kawasaki / fluctuating hydrodynamics people, this is the bounded-domain reference for CLT and LDP behavior. It deserves a serious referee. My recommendation: send it to review, and instruct the referee to enforce a complete proof of Theorem 6 -- or, if it is genuinely a re-run of DFG20, a precise theorem statement with all constants tracked.

Referee Report

4 major / 4 minor

Summary. The paper studies the generalized Dean--Kawasaki SPDE (1) on a bounded C^2 domain with Dirichlet boundary data and truncated, spatially correlated noise. It proves quantitative law-of-large-numbers estimates for a regularized equation (Propositions 4 and 5), a central limit theorem in H^{-s} for the regularized equation (Theorem 5), an L^\infty estimate and a CLT in probability for the singular equation (Theorems 6 and 7) under a joint \varepsilon-K scaling, and a large deviations principle on L^1(U\times[0,T]) with a rate function matching the zero-range-process rate function (Theorem 8). The CLT is proved only under Assumption 2, where the initial condition and the Dirichlet boundary data coincide with the same random positive constant M, so the hydrodynamic limit is the constant M. The LDP section considers general H^1(\partial U) boundary data via the weak approach of Budhiraja--Dupuis--Maroulas, relying on skeleton-equation well-posedness imported from Fehrman--Gess and the author's prior work. The paper identifies the explicit joint scaling \varepsilon K^{d+2}\to 0 for eigenfunction noise in Remark 17.

Significance. If the proof gaps are repaired, the results would be a meaningful extension of Dean--Kawasaki fluctuation theory from the torus to bounded domains with Dirichlet boundary conditions, covering fast-diffusion and porous-medium nonlinearities \Phi(\xi)=\xi^m for all m>0, including the critical square-root diffusion, and matching the zero-range-process rate function. Strengths of the manuscript include the detailed estimate chains in Theorem 5 and Propositions 4--5, the explicit joint scaling in Remark 17, and the careful discussion in Section 4.2 of why uniformity over boundary data fails. However, the singular CLT depends on an unproved L^\infty bound, and the canonical eigenfunction noise does not satisfy Assumption 1 as stated, so the paper's advertised scope is not currently established.

major comments (4)

[Section 2, Assumption 1, points 3 and 5] Assumption 1(5) is incompatible with Assumption 1(3) for any nontrivial noise. Since f_k vanishes on \partial U, the sum F_1=\sum_k f_k^2 vanishes on \partial U; a continuous function on U that is constant and vanishes on \partial U must be identically zero, forcing f_k\equiv 0 for every k. Thus the "probabilistically stationary" condition cannot be met by nonzero Dirichlet-compatible noise. In particular, Example 1 does not satisfy Assumption 1(5): for U=(0,\pi) with e_k=(2/\pi)^{1/2}\sin(kx), F_1^K(x) is a nontrivial trigonometric sum, not a constant. The theorems in Sections 3 and 4 all assume Assumption 1, so the main example and the critical square-root case are not covered as stated. Please remove or reformulate point 5, or prove that the results hold without it and state exactly where point 5 is used.
[Section 3.2, Theorem 6, equations (51)--(53)] The proof of the L^\infty bound for the singular equation is incomplete. Equation (52) is asserted to follow from (51) "as in Theorem 3.9 of [DFG20]" without showing the log-convexity/Sobolev/H\"older argument, and this is the only step that upgrades L^p control to L^\infty. The Moser iteration then starts at p_0=0 and applies (52) with p=q_1=p_0+2=2, whose right-hand side is the undefined L^0-norm of (\rho^{\varepsilon,K}-M)_-; moreover, the recursion p_k=((2+d)/d)p_{k-1}+2 does not match the exponent produced by (52), which is ((2+d)/d)(p_{k-1}+2)=p_k+4/d. Consequently the asserted bound E\|(\rho^{\varepsilon,K}-M)_-\|_{L^\infty}\le c(\varepsilon(\|F_3^K\|+\|\nabla\cdot F_2^K\|))^\gamma is not established. Since Theorem 7 uses exactly this bound to control P(S^c)+P(\tilde S^c), the central limit theorem for the singular equation rests on a gap.
[Section 4.1, Theorem 9] Theorem 9 states well-posedness of the skeleton equation, equivalence of weak and kinetic solutions, and an L^1-contraction for general H^1(\partial U) boundary data, but the proof is a one-sentence assertion that the torus result of [FG23] is adapted with arguments from [Pop25]. No details are given for the Dirichlet boundary conditions or for the general coefficients, and the statement says "under some assumptions" without listing them. Proposition 8, Theorem 10, and Proposition 15 all rely on Theorem 9, so the large deviations principle is conditional on an unverified adaptation. Please provide a complete proof or a precise reduction to the cited results that verifies every hypothesis, and clarify how Proposition 15's appeal to Theorem 21 of [FG23] is adapted to the bounded-domain setting.
[Section 3, Assumption 2, and abstract] The CLT is proved only under Assumption 2, where the initial condition and the Dirichlet boundary data coincide with the same random positive constant M. This restriction is acknowledged in Remarks 4 and 15, but the abstract presents the CLT as applying to general Dirichlet boundary conditions. Please state explicitly in the abstract and introduction that the CLT is for constant coinciding initial and boundary data, and that variable boundary data are treated only in the LDP.

minor comments (4)

[Theorem 7, statement] The phrase "Let v^{\varepsilon,K} be a weak solution to (3)" should be "Let v^{\varepsilon,K} be defined by (3)", since (3) defines v^{\varepsilon,K} in terms of the kinetic solution \rho^{\varepsilon,K} and the hydrodynamic limit \bar\rho.
[Section 3.2, equations (52)--(53)] The notation n_p conflicts with the regularization index n: in (52) and (53) n_p is used both as an exponent exponent and as the reciprocal p^{-1}. Please disambiguate these quantities.
[Section 3.2, Theorem 6] The text repeatedly writes "Mozer iteration"; this should be "Moser iteration".
[Proposition 6, uniqueness proof, equation (34) and following] The double-sum estimates in the uniqueness proof contain repeated or mismatched summation indices; the sums over j and k should be written separately with the weighting by \lambda_j^{-s} displayed consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CLT and LDP are derived from energy estimates, the Budhiraja–Dupuis–Maroulas weak approach, and published prior well-posedness; no fitted parameter or assumed conclusion drives the main results.

full rationale

The paper does not define any input in terms of its claimed output, and the two main theorems are not instances of the hypotheses. Theorem 5 is proved from Itô-formula energy estimates (Propositions 4 and 5) and a comparison with the linearized SPDE (4); Theorem 7 passes to the singular equation via the L∞ estimate of Theorem 6 and the pathwise uniqueness from [Pop25], a prior published work whose stated assumptions do not include the CLT or LDP. The LDP in Theorem 8 is proved through the BDM08 weak approach: it verifies existence of solution maps (Propositions 8 and 10), tightness of controlled SPDEs (Corollary 1), convergence to the skeleton equation (Theorem 10), and lower semicontinuity of the rate function (Proposition 15); the rate function is not imposed as the large-deviation limit. The match with zero-range-process fluctuations in Section 1.1 is a comparison with independent benchmarks [KL98, BKL95], not an input. The scaling regime in Remark 17 is a sufficient condition making the right-hand sides vanish, not a calibration. The only soft spot is the compressed Moser step in Theorem 6 around equation (52), which invokes Theorem 3.9 of [DFG20] for the improved estimate; this is a proof-detail/correctness risk, not a circular reduction, since the quantity being estimated (the L∞ negative part of ρ^{ε,K}−M) is not part of the assumptions. Overall: no fitted parameter is renamed as a prediction, and no self-citation carries the central claim by itself.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The paper is a pure theorem paper; no data are fitted. The central claim rests on inherited well-posedness from Pop25, structural assumptions on nonlinearities, and background PDE and probability tools. The most restrictive extra input is Assumption 2 for the CLT.

free parameters (1)

M (constant initial/boundary data) = M > 0, arbitrary, not estimated
Assumption 2 imposes ρ0 = boundary value = M; the CLT concerns fluctuations around this constant. It is not fitted to data but is a user-chosen parameter that limits the result.

assumptions (8)

domain assumption Assumption 1: truncated noise defined by Dirichlet eigenfunctions with bounded sums F_i^K and Dirichlet homogeneous spatial modes.
Needed for noise truncation and boundary behavior; not derived in this paper.
ad hoc to paper Assumption 2: initial and boundary data equal the same random positive constant M.
Imposed to make the hydrodynamic limit constant and energy estimates work; it is not needed for the LDP and restricts the CLT.
domain assumption Assumption 3: polynomial bounds (16)-(18) on Φ'', ν'', and auxiliary functions.
Growth and regularity conditions on nonlinearities used in Theorem 5 and in the energy estimates.
domain assumption Assumption 6: coefficient conditions on Φ, σ, ν, including σ^2 ≤ cΦ, growth and entropy conditions, from Pop25.
Inherited well-posedness assumptions for kinetic solutions of (1).
domain assumption Assumption 7: boundary data regularity for the LDP, including Φ^{-1}(\bar f) ∈ H^1(∂U) and entropy boundary terms.
Used for weak solutions of the skeleton equation and the controlled SPDE.
domain assumption Well-posedness and L1 contraction for the skeleton equation (Theorem 9).
The proof in this paper is one sentence citing an adaptation of FG23 and Pop25; this result is central to the rate-function definition and the solution maps.
standard math Background tools: Aubin-Lions-Simon compactness, Sobolev extension theorem, Weyl's law, Girsanov theorem.
These are classical tools used for tightness, regularity on C^2 domains, spectral scaling, and change of measure.
domain assumption Entropy space Ent_Φ(U) with finite entropy initial data.
Required for the initial data in the LDP; Section 5.3 gives a formal estimate motivating this space.

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Pith. "Pith review of Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions." pith.science (2026). https://pith.science/paper/PZTW4WKP

@misc{pith2026250417094,
  author       = {Pith},
  title        = {Pith review of: Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZTW4WKP}},
  note         = {Machine review of arXiv:2504.17094}
}

abstract

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general $C^2$ bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.

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Works this paper leans on

66 extracted references · 52 canonical work pages

[1]

Un th \'e oreme de compacit \'e

Jean-Pierre Aubin. Un th \'e oreme de compacit \'e . CR Acad. Sci. Paris , 256(24):5042--5044, 1963

work page 1963
[2]

Analysis and approximation of rare events

Amarjit Budhiraja and Paul Dupuis. Analysis and approximation of rare events. Representations and Weak Convergence Methods. Series Prob. Theory and Stoch. Modelling , 94:8, 2019

work page 2019
[3]

Large deviations for infinite dimensional stochastic dynamical systems

Amarjit Budhiraja, Paul Dupuis, and Vasileios Maroulas. Large deviations for infinite dimensional stochastic dynamical systems. The Annals of Probability , 36(4):1390--1420, 2008

work page 2008
[4]

The cauchy--dirichlet problem for the fast diffusion equation on bounded domains

Matteo Bonforte and Alessio Figalli. The cauchy--dirichlet problem for the fast diffusion equation on bounded domains. Nonlinear Analysis , 239:113394, 2024

work page 2024
[5]

Large deviations and transitions between equilibria for stochastic landau--lifshitz--gilbert equation

Zdzis aw Brze \'z niak, Ben Goldys, and Terence Jegaraj. Large deviations and transitions between equilibria for stochastic landau--lifshitz--gilbert equation. Archive for Rational Mechanics and Analysis , 226:497--558, 2017

work page 2017
[6]

Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations

Mostafa Bendahmane and Kenneth H Karlsen. Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations. SIAM journal on mathematical analysis , 36(2):405--422, 2004

work page 2004
[7]

Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes

Olivier Benois, Claude Kipnis, and Claudio Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic processes and their applications , 55(1):65--89, 1995

work page 1995
[8]

Large deviations for the dynamic d 2 n model

Sandra Cerrai and Arnaud Debussche. Large deviations for the dynamic d 2 n model. Applied Mathematics & Optimization , 80(1):81--102, 2019

work page 2019

Show all 66 references

[9]

Approximation of quasi-potentials and exit problems for multidimensional rde’s with noise

Sandra Cerrai and Mark Freidlin. Approximation of quasi-potentials and exit problems for multidimensional rde’s with noise. Transactions of the American Mathematical Society , 363(7):3853--3892, 2011

2011
[10]

A central limit theorem for nonlinear conservative spdes

Andrea Clini and Benjamin Fehrman. A central limit theorem for nonlinear conservative spdes. arXiv preprint arXiv:2310.19924 , 2023

2023 arXiv
[11]

Wealth concentration in systems with unbiased binary exchanges

Ben-Hur Francisco Cardoso, Jos \'e Roberto Iglesias, and Sebasti \'a n Gon c alves. Wealth concentration in systems with unbiased binary exchanges. Physica A: Statistical Mechanics and its Applications , 579:126123, 2021

2021
[12]

Central limit theorems for parabolic stochastic partial differential equations

Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu. Central limit theorems for parabolic stochastic partial differential equations. In Annales de l'Institut Henri Poincare (B) Probabilites et statistiques , volume 58, pages 1052--1077. Institut Henri Poincar \'e , 2022

2022
[13]

Porous media equations with nonlinear gradient noise and dirichlet boundary conditions

Andrea Clini. Porous media equations with nonlinear gradient noise and dirichlet boundary conditions. Stochastic Processes and their Applications , 159:428--498, 2023

2023
[14]

Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations

Gui-Qiang Chen and Beno\^it Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. In Annales de l'IHP Analyse non lin\'eaire , volume 20, pages 645--668, 2003

2003
[15]

A weak convergence approach to the theory of large deviations

Paul Dupuis and Richard S Ellis. A weak convergence approach to the theory of large deviations . John Wiley & Sons, 2011

2011
[16]

Langevin equation for the density of a system of interacting langevin processes

David S Dean. Langevin equation for the density of a system of interacting langevin processes. Journal of Physics A: Mathematical and General , 29(24):L613, 1996

1996
[17]

Conservative stochastic pde and fluctuations of the symmetric simple exclusion process

Nicolas Dirr, Benjamin Fehrman, and Benjamin Gess. Conservative stochastic pde and fluctuations of the symmetric simple exclusion process. arXiv preprint arXiv:2012.02126 , 2020

2012 arXiv
[18]

Nonlinear diffusion equations with nonlinear gradient noise

Konstantinos Dareiotis and Benjamin Gess. Nonlinear diffusion equations with nonlinear gradient noise. Electronic Journal of Probability , 25:1--43, 2020

2020
[19]

Entropy solutions for stochastic porous media equations

Konstantinos Dareiotis, M \'a t \'e Gerencs \'e r, and Benjamin Gess. Entropy solutions for stochastic porous media equations. Journal of Differential Equations , 266(6):3732--3763, 2019

2019
[20]

Weak error of dean-kawasaki equation with smooth mean-field interactions

Ana Djurdjevac, Xiaohao Ji, and Nicolas Perkowski. Weak error of dean-kawasaki equation with smooth mean-field interactions. arXiv preprint arXiv:2502.20929 , 2025

2025 arXiv
[21]

Weak error analysis for a nonlinear spde approximation of the dean-kawasaki equation

Ana Djurdjevac, Helena Kremp, and Nicolas Perkowski. Weak error analysis for a nonlinear spde approximation of the dean-kawasaki equation. arXiv preprint arXiv:2212.11714 , 2022

2022 arXiv
[22]

Stochastic equations in infinite dimensions , volume 152

Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions , volume 152. Cambridge university press, 2014

2014
[23]

Entropic and gradient flow formulations for nonlinear diffusion

Nicolas Dirr, Marios Stamatakis, and Johannes Zimmer. Entropic and gradient flow formulations for nonlinear diffusion. Journal of Mathematical Physics , 57(8):081505, 2016

2016
[24]

A central limit theorem for the stochastic wave equation with fractional noise

Francisco Delgado-Vences, David Nualart, and Guangqu Zheng. A central limit theorem for the stochastic wave equation with fractional noise. 2020

2020
[25]

Large deviation principles for first-order scalar conservation laws with stochastic forcing

Zhao Dong, Jiang-Lun Wu, Rangrang Zhang, and Tusheng Zhang. Large deviation principles for first-order scalar conservation laws with stochastic forcing. The Annals of Applied Probability , 30(1):324--367, 2020

2020
[26]

Partial differential equations , volume 19

Lawrence C Evans. Partial differential equations , volume 19. American Mathematical Society, 2022

2022
[27]

Large deviations for conservative stochastic pde and non-equilibrium fluctuations

Benjamin Fehrman and Benjamin Gess. Large deviations for conservative stochastic pde and non-equilibrium fluctuations. arXiv preprint arXiv:1910.11860 , 2019

1910 arXiv
[28]

Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise

Benjamin Fehrman and Benjamin Gess. Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise. Archive for Rational Mechanics and Analysis , 233(1):249--322, 2019

2019
[29]

Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

Benjamin Fehrman and Benjamin Gess. Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise. Journal de Math \'e matiques Pures et Appliqu \'e es , 148:221--266, 2021

2021
[30]

Non-equilibrium large deviations and parabolic-hyperbolic pde with irregular drift

Benjamin Fehrman and Benjamin Gess. Non-equilibrium large deviations and parabolic-hyperbolic pde with irregular drift. Inventiones mathematicae , 234(2):573--636, 2023

2023
[31]

Conservative stochastic pdes on the whole space

Benjamin Fehrman and Benjamin Gess. Conservative stochastic pdes on the whole space. arXiv preprint arXiv:2410.00254 , 2024

2024 arXiv
[32]

Well-posedness of the dean--kawasaki and the nonlinear dawson--watanabe equation with correlated noise

Benjamin Fehrman and Benjamin Gess. Well-posedness of the dean--kawasaki and the nonlinear dawson--watanabe equation with correlated noise. Archive for Rational Mechanics and Analysis , 248(2):20, 2024

2024
[33]

Potential techniques for boundary value problems on c1-domains

Eugene B Fabes, Max Jodeit Jr, and Nestor M Rivi \`e re. Potential techniques for boundary value problems on c1-domains. Acta mathematica , 141(1):165--186, 1978

1978
[34]

Regularity of the distance function

Robert L Foote. Regularity of the distance function. Proceedings of the American Mathematical Society , 92(1):153--155, 1984

1984
[35]

Local equilibrium for a one dimensional zero range process

Pablo Augusto Ferrari, E Presutti, and ME Vares. Local equilibrium for a one dimensional zero range process. Stochastic processes and their applications , 26:31--45, 1987

1987
[36]

Paracontrolled distributions and singular pdes

Massimiliano Gubinelli, Peter Imkeller, and Nicolas Perkowski. Paracontrolled distributions and singular pdes. In Forum of Mathematics, Pi , volume 3, page e6. Cambridge University Press, 2015

2015
[37]

A theory of regularity structures

Martin Hairer. A theory of regularity structures. Inventiones mathematicae , 198(2):269--504, 2014

2014
[38]

A central limit theorem for the stochastic heat equation

Jingyu Huang, David Nualart, and Lauri Viitasaari. A central limit theorem for the stochastic heat equation. Stochastic Processes and Their Applications , 130(12):7170--7184, 2020

2020
[39]

Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions

Kyozi Kawasaki. Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions. Physica A: Statistical Mechanics and its Applications , 208(1):35--64, 1994

1994
[40]

Scaling limits of interacting particle systems , volume 320

Claude Kipnis and Claudio Landim. Scaling limits of interacting particle systems , volume 320. Springer Science & Business Media, 1998

1998
[41]

Dean-kawasaki dynamics: ill-posedness vs

Vitalii Konarovskyi, Tobias Lehmann, and Max-K von Renesse. Dean-kawasaki dynamics: ill-posedness vs. triviality. Electronic Communications in Probability , 24:1--9, 2019

2019
[42]

On dean--kawasaki dynamics with smooth drift potential

Vitalii Konarovskyi, Tobias Lehmann, and Max von Renesse. On dean--kawasaki dynamics with smooth drift potential. Journal of Statistical Physics , 178(3):666--681, 2020

2020
[43]

Zero-range model of traffic flow

J Kaupu z s, R Mahnke, and RJ Harris. Zero-range model of traffic flow. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , 72(5):056125, 2005

2005
[44]

On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

Kenneth Hvistendahl Karlsen and Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. 2000

2000
[45]

On the uniqueness and on the smoothness of weak solutions of the navier--stokes equations

Olga Aleksandrovna Ladyzhenskaya. On the uniqueness and on the smoothness of weak solutions of the navier--stokes equations. Zapiski Nauchnykh Seminarov POMI , 5:169--185, 1967

1967
[46]

Eigenvalues of the laplacian

Julie Levandosky. Eigenvalues of the laplacian. https://web.stanford.edu/class/math220b/handouts/eigenvalues.pdf, 2003. Accessed: 06 March 2025

2003
[47]

" quelques m \'e thodes de r \'e solution des probl \`e mes aux limites non-lin \'e aires,''

J-L Lions. " quelques m \'e thodes de r \'e solution des probl \`e mes aux limites non-lin \'e aires,''. Dunod , 1969

1969
[48]

A kinetic formulation of multidimensional scalar conservation laws and related equations

P-L Lions, Benoit Perthame, and Eitan Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. Journal of the American Mathematical Society , 7(1):169--191, 1994

1994
[49]

Fully nonlinear stochastic partial differential equations

Pierre-Louis Lions and Panagiotis E Souganidis. Fully nonlinear stochastic partial differential equations. Comptes Rendus de l'Acad \'e mie des Sciences-Series I-Mathematics , 326(9):1085--1092, 1998

1998
[50]

Fully nonlinear stochastic pde with semilinear stochastic dependence

Pierre-Louis Lions and Panagiotis E Souganidis. Fully nonlinear stochastic pde with semilinear stochastic dependence. Comptes Rendus de l'Acad \'e mie des Sciences-Series I-Mathematics , 331(8):617--624, 2000

2000
[51]

Differential equations driven by rough signals

Terry J Lyons. Differential equations driven by rough signals. Revista Matem \'a tica Iberoamericana , 14(2):215--310, 1998

1998
[52]

An additive noise approximation to keller-segel-dean-kawasaki dynamics part i: Local well-posedness of paracontrolled solutions

Adrian Martini and Avi Mayorcas. An additive noise approximation to keller-segel-dean-kawasaki dynamics part i: Local well-posedness of paracontrolled solutions. arXiv preprint arXiv:2207.10711 , 2022

2022
[53]

Well-posedness of the generalised dean--kawasaki equation with correlated noise on bounded domains

Shyam Popat. Well-posedness of the generalised dean--kawasaki equation with correlated noise on bounded domains. Stochastic Processes and their Applications , 179:104503, 2025

2025
[54]

A game theoretic model of wealth distribution

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, and Nicolas Saintier. A game theoretic model of wealth distribution. Dynamic Games and Applications , 8:874--890, 2018

2018
[55]

Un teorema di unicita per le equazioni di navier-stokes

Giovanni Prodi. Un teorema di unicita per le equazioni di navier-stokes. Annali di Matematica pura ed applicata , 48:173--182, 1959

1959
[56]

Numerical approximation of partial differential equations , volume 23

Alfio Quarteroni and Alberto Valli. Numerical approximation of partial differential equations , volume 23. Springer Science & Business Media, 2008

2008
[57]

Large deviations for stochastic generalized porous media equations

Michael R \"o ckner, Feng-Yu Wang, and Liming Wu. Large deviations for stochastic generalized porous media equations. Stochastic processes and their applications , 116(12):1677--1689, 2006

2006
[58]

Continuous martingales and Brownian motion , volume 293

Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion , volume 293. Springer Science & Business Media, 2013

2013
[59]

Singular integrals on compact manifolds

Robert T Seeley. Singular integrals on compact manifolds. American Journal of Mathematics , 81(3):658--690, 1959

1959
[60]

On the interior regularity of weak solutions of the Navier-Stokes equations

James Serrin. On the interior regularity of weak solutions of the Navier-Stokes equations . Mathematics Division, Air Force Office of Scientific Research, 1961

1961
[61]

Compact sets in the space l p (o, t; b)

Jacques Simon. Compact sets in the space l p (o, t; b). Annali di Matematica pura ed applicata , 146:65--96, 1986

1986
[62]

Partial differential equations: An introduction

Walter A Strauss. Partial differential equations: An introduction . John Wiley & Sons, 2007

2007
[63]

U ber die asymptotische verteilung der eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu G \

Hermann Weyl. \"U ber die asymptotische verteilung der eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu G \"o ttingen, Mathematisch-Physikalische Klasse , 1911:110--117, 1911

1911
[64]

Dean-kawasaki equation with singular non-local interactions and correlated noise

Likun Wang, Zhengyan Wu, and Rangrang Zhang. Dean-kawasaki equation with singular non-local interactions and correlated noise. arXiv preprint arXiv:2207.12774 , 2022

2022 arXiv
[65]

Large deviations for dean-kawasaki equation with singular non-local interactions

Zhengyan Wu and Rangrang Zhang. Large deviations for dean-kawasaki equation with singular non-local interactions. arXiv preprint arXiv:2208.13142 , 2022

2022 arXiv
[66]

Large deviations for stochastic porous media equations

Rangrang Zhang. Large deviations for stochastic porous media equations. 2020

2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Un th \'e oreme de compacit \'e

Jean-Pierre Aubin. Un th \'e oreme de compacit \'e . CR Acad. Sci. Paris , 256(24):5042--5044, 1963

work page 1963

[2] [2]

Analysis and approximation of rare events

Amarjit Budhiraja and Paul Dupuis. Analysis and approximation of rare events. Representations and Weak Convergence Methods. Series Prob. Theory and Stoch. Modelling , 94:8, 2019

work page 2019

[3] [3]

Large deviations for infinite dimensional stochastic dynamical systems

Amarjit Budhiraja, Paul Dupuis, and Vasileios Maroulas. Large deviations for infinite dimensional stochastic dynamical systems. The Annals of Probability , 36(4):1390--1420, 2008

work page 2008

[4] [4]

The cauchy--dirichlet problem for the fast diffusion equation on bounded domains

Matteo Bonforte and Alessio Figalli. The cauchy--dirichlet problem for the fast diffusion equation on bounded domains. Nonlinear Analysis , 239:113394, 2024

work page 2024

[5] [5]

Large deviations and transitions between equilibria for stochastic landau--lifshitz--gilbert equation

Zdzis aw Brze \'z niak, Ben Goldys, and Terence Jegaraj. Large deviations and transitions between equilibria for stochastic landau--lifshitz--gilbert equation. Archive for Rational Mechanics and Analysis , 226:497--558, 2017

work page 2017

[6] [6]

Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations

Mostafa Bendahmane and Kenneth H Karlsen. Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations. SIAM journal on mathematical analysis , 36(2):405--422, 2004

work page 2004

[7] [7]

Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes

Olivier Benois, Claude Kipnis, and Claudio Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic processes and their applications , 55(1):65--89, 1995

work page 1995

[8] [8]

Large deviations for the dynamic d 2 n model

Sandra Cerrai and Arnaud Debussche. Large deviations for the dynamic d 2 n model. Applied Mathematics & Optimization , 80(1):81--102, 2019

work page 2019