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Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

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Pith's one-line read This paper establishes existence and uniqueness of an invariant probability measure for anisotropic degenerate parabolic-hyperbolic conservation laws driven by additive white noise.

desk verdict The existence theorem is a credible extension of Debussche-Vovelle, but the uniqueness proof breaks on a false mollification bound in Lemma 5.2. read the letter →

arxiv 1908.04879 v2 pith:UIS6JDG7 submitted 2019-08-13 math.AP math.PRnlin.CD

Gui-Qiang G. Chen , Peter H. C. Pang This is my paper · ORCID
classification math.APmath.PRnlin.CD MSC 35B4035K6537-0237A5037C4060H15
keywords invariantmeasuresstochasticconservationlawsanisotropicdegenerateparabolic-hyperbolicequationskineticformulationadditivewhitenoiseKrylov-Bogoliubovmethodcouplinglong-timebehavior
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's central claim is that a second-order conservation law with degenerate diffusion, nonlinear flux, and additive white-in-time noise settles into a unique statistical steady state: there is exactly one invariant probability measure describing its long-run behavior. The equation studied is the anisotropic degenerate parabolic-hyperbolic balance law $\partial_t u + \nabla\cdot F(u) = \nabla\cdot(A(u)\nabla u) + \sigma(x)\partial_t W$ on the torus, with the noise having zero spatial average. Existence of an invariant measure is proven under a joint nonlinearity-diffusivity condition, and uniqueness is proven under the additional condition that $F''$ and $A'$ are bounded. A sympathetic reader cares because this turns the long-time behavior of stochastic conservation laws, not just deterministic ones, into a well-defined statistical question.

What carries the argument

The central object is the kinetic formulation of the equation, in which the indicator function $\chi(\xi,u)$ is transported in phase space, so the nonlinear flux becomes a linear first-order operator in $\xi$ and $x$ at the cost of adding kinetic measures that carry the dissipation. The non-degeneracy condition (4.2) quantifies how much velocity averaging regularizes oscillations in $\xi$, and the regularizing operators $\gamma(-\Delta)^\alpha + \theta I$ inserted into the kinetic equation make the semigroup estimates possible. These pieces together yield the compactness in $W^{s,q}$ needed for the Krylov-Bogoliubov existence argument, while the $L^1$-contraction property of the original equation is what lets the coupling stopping times force uniqueness.

What would settle it

Take initial data $u_0$ that concentrate unit mass in a tiny subset of volume $\delta^d$ with $\delta$ much smaller than $\varepsilon$, and compute the minimum $L^2$ norm among all functions within $L^1$ distance $\varepsilon/8$ of $u_0$; that minimum is about $\delta^{-d/2}$, which far exceeds the claimed $C\hat\kappa\,\varepsilon^{-d/2}$ bound and would invalidate Step 1 of Lemma 5.2.

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Extended reading notes

Core claim

On the paper's own terms: for the stochastic anisotropic degenerate parabolic-hyperbolic equation with additive noise $\sigma(x)\partial_t W$, the solution semigroup admits at least one invariant probability measure, and under a boundedness condition this measure is unique. The proof treats the unknown through its kinetic formulation, decomposes the solution into deterministic, martingale, and Itô-correction parts, and uses regularizing operators to obtain compactness estimates that feed into the Krylov-Bogoliubov argument. Uniqueness is obtained by a coupling argument: two solutions starting from different data enter the same small ball infinitely often with probability one, and the $L^1$-contraction property of the equation then forces their $L^1$ distance to tend to zero almost surely.

Load-bearing premise

The load-bearing premise is that every $L^1$ initial datum inside the recurrence ball can be mollified into a nearby function whose $L^2$ norm is bounded by a constant times the radius times $\varepsilon^{-d/2}$; the uniqueness theorem's recurrence estimate collapses if this approximation bound cannot be enforced.

Editorial extensions

If this is right

  • If the theorems are right, any two $L^1$ initial data give solutions whose $L^1$ distance tends to zero almost surely, so the initial state is asymptotically forgotten.
  • The unique invariant measure is ergodic, so statistics along a single long trajectory reproduce the measure's long-run statistics.
  • The existence theorem covers fully degenerate hyperbolic cases, not only uniformly parabolic equations, as long as the nonlinearity-diffusivity condition (4.2) holds.
  • Taking $A=0$ recovers the first-order scalar balance law case with additive spatially dependent noise, with uniqueness under bounded $F''$.
  • The zero-spatial-average assumption on $\sigma$ is used throughout, so the result is tied to noise that does not directly inject mass into the conserved quantity.

Reading between the lines

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

  • Editorial: The uniqueness proof as written relies on Step 1 of Lemma 5.2 being able to replace any $L^1$ datum of size at most $2\hat\kappa$ by a mollified datum close in $L^1$ with $L^2$ norm $O(\varepsilon^{-d/2})$; a concentrated initial datum violates this bound, leaving a gap in the proof of uniqueness as written.
  • Editorial: Because $L^1$ contraction is the only pathwise stability input, the same argument cannot mechanically extend to multiplicative noise; the natural test case is noise with a root, where the root is a fixed point and one should check whether more than one invariant measure appears.
  • Editorial: The recurrence probability depends only on a Brownian increment staying small in $W^{1,\infty}$ over a fixed time window, so any noise with the same positive small-ball probability would likely inherit the coupling argument.
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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

3 major / 5 minor

Summary. The paper proposes an existence and uniqueness theory of invariant measures for stochastic anisotropic degenerate parabolic-hyperbolic conservation laws of second order driven by additive spatial white noise, equation (4.1). Section 4 develops an existence argument via the kinetic formulation, a regularized mild formulation, a four-term decomposition into u0, u♭, M1, M2, kernel estimates, and compactness in Sobolev spaces, followed by an invocation of the Krylov-Bogoliubov machinery. Section 5 aims to prove uniqueness through a coupling/recurrence argument: Lemma 5.1 provides almost-sure finite-time entry into a ball, Lemma 5.2 gives temporal L1 averaging bounds under small noise, and Section 5.3 concludes by combining L1 contraction with a Borel-Cantelli argument. The paper also contains a broad survey of invariant measures for stochastic conservation laws, Burgers equation, Navier-Stokes equations, the KPP equation, and large deviations.

Significance. If established, Theorems 4.1 and 5.1 would extend the invariant-measure results of Debussche-Vovelle [35] from first-order stochastic balance laws to second-order degenerate parabolic-hyperbolic equations, a nontrivial and interesting step. The architectural ideas in Section 4, including the regularized semigroup, the four-term decomposition, and the use of kernel estimates to handle the stochastic and kinetic-measure terms, are potentially reusable and clearly presented. The survey part is broad and useful. However, the uniqueness proof in Theorem 5.1 rests on a false uniform mollification estimate in Lemma 5.2, and the existence proof in Section 4 never verifies the Feller property that the invoked Krylov-Bogoliubov theorem requires. These are load-bearing defects, so the central claims are not established in the present manuscript.

major comments (3)
  1. [§5.2, Lemma 5.2, Step 1] Step 1 claims that for every u0 with ||u0||_{L1} ≤ 2κ̂ there exists a mollified approximation ˜u0 satisfying ||u0 - ˜u0||_{L1} ≤ ε/8 and ||˜u0||_{L2} ≤ C κ̂ ε^{-d/2}. This statement is false. Take u0 = δ^{-d} 1_{B_δ} on T^d with ||u0||_{L1}=1 and δ << ε. For any f with ||u0 - f||_{L1} ≤ ε/8, the mass of f on B_δ is at least 1 - ε/8, so ||f||_{L2} ≥ (1 - ε/8)δ^{-d/2}. For δ sufficiently small this exceeds the claimed C κ̂ ε^{-d/2}. Thus no such uniform bound can hold. This estimate is essential in Step 8, where it produces the bound |m_v + N_v| ≤ C e^{Cκ̃t}(||˜u0||_{L2}^2 + 1) ≤ C e^{Cκ̃t}(κ̂^2 ε^{-d} + 1), which feeds directly into (5.16) and the final averaging estimate in Step 9. The recurrence and uniqueness proof in §5.3 therefore collapse.
  2. [§4.5, Completion of the existence proof] The proof of Theorem 4.1 invokes the Krylov-Bogoliubov mechanism of §2.2 without verifying the Feller property of the solution semigroup. Theorem 2.1 explicitly requires that P_s maps C(X) into C(X); the tightened compactness estimate (4.21) alone is insufficient for the argument as written. The paper mentions in §2.2 a replacement technique from [21] that avoids the Feller condition, but that technique is not developed or applied here. Consequently, Theorem 4.1 is incomplete as stated.
  3. [§5.2, Lemma 5.3] The almost-sure L1 contraction and well-posedness stated in Lemma 5.3 are imported from the companion paper [12] without proof. The final uniqueness argument in §5.3 uses this contraction to transfer averaging estimates from the regularized solutions ˜u1, ˜u2 to the original solutions u1, u2, so the statement is load-bearing. The manuscript should either reproduce the proof or explicitly state that Theorem 5.1 is conditional on the results of [12]; as it stands, the dependency is not transparent to the reader.
minor comments (5)
  1. [§4.3, Lemma 4.1] In the definition D0 := ||σ^2||_{L∞(T)}, the domain should presumably be T^d, not T.
  2. [§5.2, Step 9] In the paragraph beginning 'For α < 1/4', the symbol C(ρ,θ) appears where C4(γ,θ) seems intended; this is a typographical error.
  3. [§5.2, Step 1] The notation for the averaged integral is introduced as 'the symbol ffl', but the displayed formulas in Lemma 5.2 use a different typesetting; the intended meaning is clear but the typography should be unified.
  4. [§4.1] The definition of the Sobolev regularity exponent (1-α)κ + α in (4.15) is not explicitly tied to the embedding constants later; a brief justification of the exponent's range would improve readability.
  5. [§3.2] In the kinetic formulation (3.6), the term ∂_ξ(m_u + n_u - p_u) is introduced formally; the footnote-like explanation of the Itô correction p_u is helpful but would benefit from a precise reference to [12] for the rigorous limiting procedure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant measure theorems are derived from kinetic estimates and an external coupling method; the flagged mollification gap is a correctness issue, not circularity.

full rationale

The invariant-measure results are not derived from the target claims. Theorem 4.1 is proved by a Krylov-Bogoliubov compactness argument built on velocity averaging and kernel estimates, adapting the external method of Debussche-Vovelle [35]; the non-degeneracy condition (4.2) enters as an assumption, not as a disguised conclusion. Theorem 5.1 is proved by a coupling/recurrence argument using the almost-sure L1 contraction of stochastic kinetic solutions. That contraction is imported from the authors' companion paper [12], which is self-citation; however, [12] establishes well-posedness and contraction under assumptions that do not include invariant-measure uniqueness, so the citation is real independent evidence rather than a circular load-bearing step. The proof does not fit any parameter to the invariant measure, rename a known result, or invoke a self-authored uniqueness theorem to forbid alternatives. The analytic gap flagged by the reader, namely Lemma 5.2, Step 1, the claimed uniform mollified L2 bound ||u0_tilde||_{L2} <= C kappa epsilon^{-d/2}, is a correctness concern, not a circularity: if true, it is an input estimate; if false, it undermines the proof but does not make the conclusion an input. No step in the derivation reduces to Eq. X = Eq. Y by construction, and no fitted quantity is renamed as a prediction. Therefore the paper has no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theorems are conditional on the kinetic solution theory from the authors' companion paper [12] and on the non-degeneracy hypotheses. The proof also uses standard probabilistic compactness theorems. No new physical entities are introduced.

free parameters (1)
  • Auxiliary regularization constants gamma and theta = chosen in the proof (Section 4 and Section 5.2 Step 9)
    Introduced ad hoc to make the kernel estimates (4.19) integrable and small; the theorems are stated independent of their values, so they are proof parameters, not physical fitted parameters.
assumptions (4)
  • standard math Prohorov theorem, Krylov-Bogoliubov theorem, Krein-Milman theorem, Gronwall inequality
    Standard probabilistic and analytic theorems invoked without proof.
  • domain assumption Well-posedness and pathwise L1 contraction for kinetic solutions of (3.5), taken from Chen-Pang [12]
    The paper relies on the authors' companion paper for the existence, uniqueness, and L1 contraction of kinetic solutions; no proof is given in this arXiv version.
  • domain assumption Feller (or suitable continuity) property of the solution semigroup enabling Krylov-Bogoliubov
    The existence proof in Section 4.5 invokes Krylov-Bogoliubov, which requires the Feller property; the paper does not prove it here and does not cite a specific theorem establishing it for (4.1).
  • domain assumption The nonlinearity-diffusivity condition (4.2) and growth/boundedness conditions (4.3)/(5.1) hold for F and A
    These are hypotheses of Theorems 4.1 and 5.1, not derived in the paper.

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Pith. "Pith review of Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing." pith.science (2026). https://pith.science/paper/UIS6JDG7

@misc{pith2026190804879,
  author       = {Pith},
  title        = {Pith review of: Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIS6JDG7}},
  note         = {Machine review of arXiv:1908.04879}
}
read the original abstract

Some recent developments in the analysis of long-time behaviors of stochastic solutions of nonlinear conservation laws driven by stochastic forcing are surveyed. The existence and uniqueness of invariant measures are established for anisotropic degenerate parabolic-hyperbolic conservation laws of second-order driven by white noises. Some further developments, problems, and challenges in this direction are also discussed.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

102 extracted references · 80 canonical work pages

  1. [12]

    Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing

    G.-Q. Chen and P. Pang. On nonlinear anisotropic degene rate parabolic-hyperbolic equations with stochastic forcing. Preprint arXiv:1903.02693, March 201 9

    work page Pith review arXiv 1903

  2. [35]

    Debussche and J

    A. Debussche and J. Vovelle. Invariant measure of scala r first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields , 163(3) (2015), 575–611

    work page 2015

  3. [21]

    Invariant measures for the stochastic one-dimensional compressible Navier-Stokes equations

    M. Coti-Zelati, N. Glatt-Holtz, and K. Trivisa. Invari ant measures for the stochastic one-dimensional compressible Navier-Stokes equations. arXiv:1802.04000v1, 2018

    work page Pith review arXiv 2018

  4. [1]

    L. Arnold. Random Dynamical Systems . Springer-Verlag: Berlin Heidelberg, 1998

    1998

  5. [2]

    Barr´ e, C

    J. Barr´ e, C. Bernardin, and R. Chertrite. Density large deviations for multidimensional stochastic hyperbolic conservation laws. J. Stat. Phys. 170(3) (2017), 466 – 491

    2017

  6. [3]

    Bauzet, G

    C. Bauzet, G. Vallet, and P. Wittbold. The Cauchy problem for conservation laws with a multiplicative stochastic perturbation. J. Hyper. Diff. Eq. 9(4) (2012), 661–709

    2012

  7. [4]

    Berg´ e and B

    B. Berg´ e and B. Saussereau. On the long-time behavior of a class of parabolic SPDEs: Monotonicity methods and exchange of stability. ESAIM: Probab. Stat. 9 (2005), 254–276

    2005

  8. [5]

    J. Birkens. Sufficient conditions for the eventual strong Feller property for degenerate stochastic evo- lutions. J. Math. Anal. Appl. 379(2) (2011), 469–481

    2011

Show all 102 references
  1. [6]

    Bouchut and L

    F. Bouchut and L. Desvillettes. Averaging lemmas withou t time Fourier transform and application to discretized kinetic equations. Proc. Royal Soc. Edin. Sec. A , 129 (1999), 19–36

    1999

  2. [7]

    Bricmont, A

    J. Bricmont, A. Kupiainen, and R. Lefevere. Ergodicity o f the 2D Navier-Stokes equations with random forcing. Comm. Math. Phys. 224(1) (2001), 65–81

    2001

  3. [8]

    Bricmont, A.Kupiainen, and R

    J. Bricmont, A.Kupiainen, and R. Lefevere. Exponential mixing of the 2D Navier-Stokes dynamics. Comm. Math. Phys. 224(1) (2002), 87–132

    2002

  4. [9]

    G.-Q. Chen, Q. Ding, and K. Karlsen. On nonlinear stochas tic balance laws. Arch. Ration. Mech. Anal. 204(3) (2012), 707–743

    2012

  5. [10]

    Chen and H

    G.-Q. Chen and H. Frid. Large time behavior of entropy so lutions in L∞ for multidimensional conser- vation laws. In: Advances in Nonlinear PDEs and Related Areas , 28–44, World Scientific, 1998

    1998

  6. [11]

    Chen and Y.-G

    G.-Q. Chen and Y.-G. Lu. A study of approaches to applyin g the theory of compensated compactness. Chinese Sci. Bull. 34 (1989), 15–19

    1989

  7. [13]

    Chen and B

    G.-Q. Chen and B. Perthame. Well-posedness for non-iso tropic degenerate parabolic-hyperbolic equa- tions. Ann. l’I.H.P. Anal. Non-Lin´ eaires, 20(4) (2003), 645–668

    2003

  8. [14]

    Chen and B

    G.-Q. Chen and B. Perthame. Large-time behavior of peri odic entropy solutions to anisotropic degen- erate parabolic-hyperbolic equations. Proc. A.M.S., 137(9) (2009), 3003–3011. 36 GUI-QIANG G. CHEN PETER H.C. PANG

    2009

  9. [15]

    Chueshov and P

    I. Chueshov and P. Vuillermot. On the large-time dynami cs of a class of parabolic equations subjected homogeneous white noise: Stratonovitch’s case. C. R. Acad. Sci. Paris , 323(1) (1996), 29–33

    1996

  10. [16]

    Chueshov and P

    I. Chueshov and P. Vuillermot. On the large-time dynami cs of a class of random parabolic equations. C. R. Acad. Sci. Paris , 322(1) (1996), 1181–1186

    1996

  11. [17]

    Chueshov and P

    I. Chueshov and P. Vuillermot. Long-time behavior of so lutions to a class of quasilinear parabolic equations with random coefficients. Ann. I.H.P. Anal. Non-Lin´ eaires, 15(2) (1998), 191–232

    1998

  12. [18]

    Chueshov and P

    I. Chueshov and P. Vuillermot. Long-time behavior of so lutions to a class of stochastic parabolic equations with homogeneous white noise: Straonovitch’s ca se. Probab. Theory Related Fields , 112(2) (1998), 149–202

    1998

  13. [19]

    Chueshov and P

    I. Chueshov and P. Vuillermot. On the large-time dynami cs of a class of parabolic equations subjected to homogeneous white noise: Itˆ o’s case. C. R. Acad. Sci. Paris , 326(1) (1998), 1299–1304

    1998

  14. [20]

    Chueshov and P

    I. Chueshov and P. Vuillermot. Long-time behavior of so lutions to a class of stochastic parabolic equations with homogeneous white noise: Itˆ o’s case. Stoch. Anal. Appl. 18(4) (2000), 581–615

    2000

  15. [22]

    H. Crauel. Markov measures for random dynamical system s. Stochastics and Stochastic Reports, 37(3) (1991), 153–173

    1991

  16. [23]

    H. Crauel. Random Probability Measures on Polish Spaces . Taylor and Francis: London, 2002

    2002

  17. [24]

    Crauel, A

    H. Crauel, A. Debussche, and F. Flandoli. Random attrac tors. J. Dynam. Diff. Eq. 9(2) (1997), 307–341

    1997

  18. [25]

    Crauel and F

    H. Crauel and F. Flandoli. Attractors for random dynami cal systems. Probab. Theory Relat. Fields , 100(3) (1994), 365–393

    1994

  19. [26]

    Da Prato and A

    G. Da Prato and A. Debussche. Two-dimensional Navier-S tokes equations driven by a space-time white noise. J. Funct. Anal. 196(1) (2002), 180–210

    2002

  20. [27]

    Da Prato and A

    G. Da Prato and A. Debussche. Ergodicity for the 3D stoch astic Navier-Stokes equations. J. Math. Pures Appl. 82(8) (2003), 877–947

    2003

  21. [28]

    Da Prato, A

    G. Da Prato, A. Debussche, and R. Temam. Stochastic Burg er’s equation. No.D.E.A. 1 (1994), 389– 402

    1994

  22. [29]

    Da Prato, F

    G. Da Prato, F. Flandoli, E. Priola, and M. R¨ ockner. Str ong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurab le drift. Ann. Prob. 41(5) (2013), 3306– 3344

    2013

  23. [30]

    Da Prato and J

    G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions . Cambridge University Press: Cambridge, 2008

    2008

  24. [31]

    Debussche

    A. Debussche. On the finite dimensionality of random att ractors. Stoch. Anal. Appl. 15(4) (1997), 473–492

    1997

  25. [32]

    Debussche

    A. Debussche. Hausdorff dimension of a random invariant set. J. Math. Pures Appl. 77(10) (1998), 967–988

    1998

  26. [33]

    Debussche, M

    A. Debussche, M. Hofmanov´ a, and J. Vovelle. Degenerate parabolic stochastic partial differential equa- tions: Quasilinear case. Ann. Probab. 44(3) (2016), 1916–1955

    2016

  27. [34]

    Debussche and J

    A. Debussche and J. Vovelle. Scalar conservation laws w ith stochastic forcing. J. Funct. Anal. 259(4) (2010), 1014–1042

    2010

  28. [36]

    Dembo, and O

    A. Dembo, and O. Zeitouni. Large Deviations Techniques and Applications. New York, Springer, 2nd Ed., 1997

    1997

  29. [37]

    Delarue, F

    F. Delarue, F. Flandoli, and D. Vincenzi. Noise prevent s collapse of Vlasov-Poisson point charges. Comm. Pure Appl. Math. 67(10) (2014), 1700–1736

    2014

  30. [38]

    Deuschel, and D

    J.-D. Deuschel, and D. Stroock. Large Deviations. Acad emic Press: London 1989

    1989

  31. [39]

    W. Doeblin. ´El´ ements d’une th´ eorie g´ en´ erale des cha ˆ ınes simples constantes de Markoff. Ann. Sci. ´Ecole Normale Superieur , 57 (1940), 61–111

    1940

  32. [40]

    Doeblin and R

    W. Doeblin and R. Fortet. Sur le cha ˆ ınes ` a liaisons com pl` ete. Bull. Soc. Math. France , 65 (1937), 132–148

    1937

  33. [41]

    Dong, J.-L

    Z. Dong, J.-L. Wu, R.-R. Zhang, and T.-S. Zhang. Large de viation principles for first-order scalar conservation laws with stochastic forcing. arXiv:1806.02955v1, 2018. INV ARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LA WS 3 7

    2018 arXiv

  34. [42]

    J. Doob. Asymptotic property of Markoff transition prob ability. Trans. Amer. Math. Soc. , 64(3) (1948), 393–421

    1948

  35. [43]

    Dupuis, R

    P. Dupuis, R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviation s. New York,Wiley, 1997

    1997

  36. [44]

    W. E, K. Khanin, A Mazel, and Ya. Sinai. Probability dist ribution functions for the random forced Burgers equation. Phys. Rev. Lett. 78(10) (1997), 1904–1907

    1997

  37. [45]

    W. E, K. Khanin, A Mazel, and Ya. Sinai. Invariant measur es for Burgers equation with stochastic forcing. Ann. Math. 151(3) (2000), 877–960

    2000

  38. [46]

    W. E, J. Mattingly, and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys. 224(1) (2001), 83 – 106

    2001

  39. [47]

    Fedrizzi, F

    E. Fedrizzi, F. Flandoli, E. Priola, and J. Vovelle. Reg ularity of stochastic kinetic equations. Electron. J. Probab., 22(48) (2017), 1–42

    2017

  40. [48]

    Fehrman and B

    B. Fehrman and B. Gess, Well-posedness of nonlinear diff usion equations with nonlinear, conservative noise. Preprint arXiv:1712.05775, 2019

    2019 arXiv

  41. [49]

    Feng and D

    J. Feng and D. Nualart. Stochastic scalar conservation laws. J. Funct. Anal. 255(2) (2008), 313–373

    2008

  42. [50]

    Flandoli

    F. Flandoli. Stochastic flows and Lyapunov exponents fo r abstract stochastic PDEs of parabolic type. In: Lyapunov Exponents Proceedings (L. Arnold, H. Crauel, J.-P. Eckamann eds. ), LNM 1486 (1991), 196–205

    1991

  43. [51]

    Flandoli

    F. Flandoli. Dissipativity and invariant measures for stochastic Navier-Stokes equations. No.D.E.A. 1(4) (1994), 403–423

    1994

  44. [52]

    Flandoli and D

    F. Flandoli and D. Gatarek. Martingale and stationary s olutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields , 102(3) (1995), 367–391

    1995

  45. [53]

    Flandoli, M

    F. Flandoli, M. Gubinelli, and E. Priola. Well-posedne ss of the transport equation by stochastic per- turbation. Invent. Math. 180(1) (2010), 1–53

    2010

  46. [54]

    Flandoli and B

    F. Flandoli and B. Maslowski. Ergodicity of the 2-D Navi er-Stokes equations under random perturba- tions. Comm. Math. Phys. 172(1) (1995), 119–141

    1995

  47. [55]

    Regularity Theory and Stochastic Flows for Parabolic SPDEs

    Franco Flandoli. Regularity Theory and Stochastic Flows for Parabolic SPDEs . Gordon and Breach Science Publishers: Singapore, 1995

    1995

  48. [56]

    F¨ oldes, S

    J. F¨ oldes, S. Friedlander, N. Glatt-Holtz, and G. Rich ards. Asymptotic analysis for randomly forced MHD. arXiv:1604.06352, 2016

    2016 arXiv

  49. [57]

    F¨ oldes, N

    J. F¨ oldes, N. Glatt-Holtz, G. Richards, and J.P. White head. Ergodicity in randomly forced Rayleigh- B´ enard convection.arXiv:1511.01247, 2015

    2015 arXiv

  50. [58]

    Freidlin, and A.D

    M.I. Freidlin, and A.D. Wentzell. Random Perturbation of Dynamical Systems . Springer: New York, 2nd Ed., 1998

    1998

  51. [59]

    Gess and P

    B. Gess and P. Souganidis. Long-time behavior, invaria nt measures and regularizing effects for stochas- tic scalar conservation laws. arXiv:1411.3939, 2014

    2014 arXiv

  52. [60]

    Gess and P

    B. Gess and P. Souganidis. Stochastic non-isotropic de generate parabolic-hyperbolic equations. arXiv:1611.01303, 2016

    2016 arXiv

  53. [61]

    Giga and T

    Y. Giga and T. Miyakawa. A kinetic construction of globa l solutions of first order quasilinear equations. Duke Math. J. 50(2) (1989), 505–515

    1989

  54. [62]

    Gy¨ ongy and D

    I. Gy¨ ongy and D. Nualart. On the stochastic Burgers equ ation in the real line. Ann. Probab. 27(2) (1999), 782–802

    1999

  55. [63]

    Hairer and J

    M. Hairer and J. Mattingly. Ergodicity of the 2D Navier- Stokes equations with degenerate stochastic forcing. Ann. Math. 164 (2006), 993–1032

    2006

  56. [64]

    Hairer and J

    M. Hairer and J. Mattingly. Spectral gaps in Wasserstei n distances and the 2D stochastic Navier-Stokes equations. Ann. Probab. 36(6) (2008), 2050–2091

    2008

  57. [65]

    Hairer and J

    M. Hairer and J. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electr. J. Probab. 16(23) (2011), 658–738

    2011

  58. [66]

    Hairer and J

    M. Hairer and J. Voss. Approximations to the stochastic Burgers equation. J. Nonlin. Sci. 21 (6) (2011), 897 – 920

    2011

  59. [67]

    Hofmanov´ a

    M. Hofmanov´ a. Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl. 123(12) (2013), 4294–4336

    2013

  60. [68]

    den Hollander

    F. den Hollander. Large Deviations. Fields Institute Monographs, A.M.S.: Providence, RJ, 2000 . 38 GUI-QIANG G. CHEN PETER H.C. PANG

    2000

  61. [69]

    Y. Le Jan. ´Equilibre statistique pour les produits de diff´ eomorphism es al´ eatoires ind´ ependants.Ann. l’I.H.P. Probab. Stat. 23(1) (1987), 111–120

    1987

  62. [70]

    K. H. Karlsen and E. Storrøsten. On stochastic conserva tion laws and Malliavin calculus. arXiv:1507.05518v2, 2015

    2015 arXiv

  63. [71]

    Khas’minskii

    R. Khas’minskii. Ergodic properties of recurrent diffu sion processes and stabilization of the solutions to the Cauchy problem for parabolic equations. Theory Probab. Appl. 5(2) (1960), 179–196

    1960

  64. [72]

    Krylov and M

    N. Krylov and M. R¨ ockner. Strong solutions of stochast ic equations with singular time dependent drift Probab. Theory Relat. Fields , 131(2) (2005), 154 – 196

    2005

  65. [73]

    Kuksin and A

    S. Kuksin and A. Shirikyan. Randomly Forced Nonlinear PDEs and Statistical Hydrodynam ics in 2 Space Dimensions. European Mathematical Society: Z¨ urich, 2006

    2006

  66. [74]

    Kuksin and A

    S. Kuksin and A. Shirikyan. Mathematics of Two-Dimensional Turbulence. Cambridge University Press: Cambridge, 2002

    2002

  67. [75]

    Ledrappier

    F. Ledrappier. Positivity of the exponent for stationa ry sequences of matrices. In: Lyapunov Exponents Proceedings (L. Arnold and V. Wihstutz, eds. ) LNM 1186 (1986), 56–73

    1986

  68. [76]

    Le` on, D

    J. Le` on, D. Nualart, and R. Pettersson The stochastic B urgers equation: Finite moments and smooth- ness of the density. Infinite Dimensional Analysis , 3 (2000), 363 – 385

    2000

  69. [77]

    Lindvall

    T. Lindvall. Lectures on the Coupling Method . John Wiley & Sons: New York, 1992

    1992

  70. [78]

    Lions, B

    P.-L. Lions, B. Perthame, and P. Souganidis. Scalar con servation laws with rough (stochastic) fluxes. Stochastic PDEs : Anal. Comput. 1 (2013), 664–686

    2013

  71. [79]

    Lions, B

    P.-L. Lions, B. Perthame, and P. Souganidis. Scalar con servation laws with rough (stochastic) fluxes: the spatially dependent case. Stochastic PDEs : Anal. Comput. 2(4) (2014), 517–538

    2014

  72. [80]

    Lions, B

    P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formu lation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7(1) (1994), 169–191

    1994

  73. [81]

    Lions, B

    P.-L. Lions, B. Perthame, and E. Tadmor. Kinetic formul ation of the isentropic gas dynamics and p-systems. Comm. Math. Phys. 163 (1994), 415–431

    1994

  74. [82]

    A. Majda. Introduction to Turbulent Dynamical Systems in Complex Sys tems. Springer: Cham, 2016

    2016

  75. [83]

    Majda, and X.-T

    A. Majda, and X.-T. Tong Ergodicity of truncated stocha stic Navier Stokes with deterministic forcing and dispersion. J. Nonlin. Sci. 26 (5) (2016), 1483–1506

    2016

  76. [84]

    Majda and X.-M

    A. Majda and X.-M. Wang. Nonlinear Dynamics and Statistical Theories for Basic Geop hysical Flows . Cambridge University Press: Cambridge, 2006

    2006

  77. [85]

    M. Mariani. Large deviations principles for stochasti c scalar conservation laws. Probab. Theory Relat. Fields 147 (2010), 607 – 648

    2010

  78. [86]

    Maslowski and J

    B. Maslowski and J. Seidler. Invariant measures for non linear SPDE’s: uniqueness and stability. Archivum Mathematicum, 34(1) (1998), 153–172

    1998

  79. [87]

    Masmoudi and L.-S

    N. Masmoudi and L.-S. Young. Ergodic theory of infinite d imensional systems with applications to dissipative parabolic PDEs. Comm. Math. Phys. 227(3) (2002), 461–481

    2002

  80. [88]

    J. C. Mattingly. Exponential convergence for the stoch astically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230(3) (2002), 421–462

    2002

  81. [89]

    J. C. Mattingly. Recent progress for the stochastic Nav ier-Stokes equations. Journe´ e´EDP. (2003), Exp. No. 11, 52 p., doi: 10.5802/jedp.625 http://www.numdam.or g/article/JEDP 2003 A11 0.pdf

    2003 doi

  82. [90]

    Perthame

    B. Perthame. Kinetic Formulation of Conservation Laws . Oxford University Press: Oxford, 2002

    2002

  83. [91]

    Peszat and J

    S. Peszat and J. Zabczyk. Strong Feller property and irr educibility for diffusions on Hilbert spaces. Ann. Probab. 23(1) (1995), 157–172

    1995

  84. [92]

    Peszat and J

    S. Peszat and J. Zabczyk. Stochastic Partial Differential Equations with L´ evy Noise: An Evolution Equation Approach. Cambridge University Press: Cambridge, 2007

    2007

  85. [93]

    R¨ ockner and X.-C

    M. R¨ ockner and X.-C. Zhang. Stochastic tamed 3D Navier -Stokes equations: existence, uniqueness and ergodicity. Prob. Theory Relat. Fields , 145 (2009), 211 – 267

    2009

  86. [94]

    Romito and L.-H

    M. Romito and L.-H. Xu. Ergodicity of the 3D stochastic N avier-Stokes equations driven by mildly degenerate noise. Stoc. Proc. Appl. 121 (2011), 673 – 700

    2011

  87. [95]

    Schmalfuss

    B. Schmalfuss. The random attractor of the stochastic L orenz system. ZAMP 48(6) (1997), 951 – 975

    1997

  88. [96]

    Ya. Sinai. Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys. 64(1) (1991), 1–12. INV ARIANT MEASURES FOR NONLINEAR STOCHASTIC BALANCE LA WS 3 9

    1991

  89. [97]

    Tadmor and T

    E. Tadmor and T. Tao. Velocity averaging, kinetic formu lations and regularizing effects in quasi-linear pdes. Comm. Pure Appl. Math. 60(10) (2007), 1488–1521

    2007

  90. [98]

    R. Temam. Infinite Dimensional Dynamical Systems in Mechanics and Phy sics. Springer: New York, 1997

    1997

  91. [99]

    Varadhan

    S.R.S. Varadhan. Large Deviations. Courant Lecture Notes, Amer. Math. Soc., 2016

    2016

  92. [100]

    Veretennikov

    A. Veretennikov. On strong solutions and explicit for mulas for solutions of stochastic integral equa- tions. Math. USSR Sbornik , 39(3) (1981)

    1981

  93. [101]

    C. Villani. Optimal Transport, Grundlehren der mathematischen Wissenschaften 338, Spri nger, 2009

    2009

  94. [102]

    Zhu and X

    R. Zhu and X. Zhu. Three-dimensional Navier-Stokes eq uations driven by space-time white noise. J. Diff. Eq. 259(9) (2015), 4443–4508. Gui-Qiang G. Chen, Mathematical Institute, University of O xford, Oxford, OX2 6GG, UK E-mail address : chengq@maths.ox.ac.uk Peter P ang, Mathe...

    2015

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