pith. sign in

arxiv: 2412.01381 · v5 · pith:DPZGI5MSnew · submitted 2024-12-02 · 🧮 math.PR · math.AP· math.DS· math.FA

Ergodicity and mixing for locally monotone stochastic evolution equations

Gerardo Barrera , Jonas M. T\"olle This is my paper

Pith reviewed 2026-05-23 08:36 UTC · model grok-4.3

classification 🧮 math.PR math.APmath.DSmath.FA
keywords stochastic evolution equationsergodicityinvariant measuresmixing timeslocally monotone driftWasserstein distanceNavier-Stokes equations
0
0 comments X

The pith

Stochastic evolution equations with locally monotone drift and degenerate noise possess a unique invariant measure and converge exponentially to it.

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

The paper establishes general conditions under which stochastic evolution equations driven by additive Wiener noise admit a unique invariant probability measure. This holds for equations posed in variational form even when the noise is degenerate, as long as the drift satisfies local monotonicity together with growth and coercivity. The associated Markov semigroup is shown to be Feller and exponentially ergodic, with explicit quantitative bounds on mixing times in the 2-Wasserstein metric. The results cover concrete examples such as the 2D Navier-Stokes equations and the 1D Burgers equation on possibly unbounded domains.

Core claim

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the e-property of the semigroup. Furthermore, we provide quantitative upper bounds for the 2-Wasserstein ε-mixing times.

What carries the argument

The local monotonicity condition on the drift coefficient (together with growth and coercivity) in a variational formulation on a Gelfand triple, which closes the estimates yielding uniqueness of the invariant measure and exponential ergodicity of the Markov semigroup.

If this is right

  • The Markov semigroup generated by the equation is Feller and exponentially ergodic.
  • Solutions satisfy improved moment estimates.
  • The semigroup satisfies the e-property.
  • Quantitative upper bounds hold for the 2-Wasserstein ε-mixing times.
  • The conclusions apply to the stochastic incompressible 2D Navier-Stokes equations, shear thickening power-law fluid equations, the stochastic heat equation, and the 1D stochastic Burgers equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Verification of local monotonicity for an additional equation would immediately imply uniqueness of its long-time statistical behavior independent of the starting point.
  • The explicit mixing-time bounds supply a concrete criterion for deciding when long-time numerical simulations have reached statistical equilibrium.
  • The allowance for degenerate noise indicates that full-dimensional forcing is not required for ergodicity once local monotonicity is present.

Load-bearing premise

The drift coefficient satisfies a local monotonicity condition together with suitable growth and coercivity that is compatible with the variational formulation on the chosen Gelfand triple.

What would settle it

An explicit stochastic evolution equation obeying local monotonicity, growth, and coercivity for which either no unique invariant measure exists or the 2-Wasserstein mixing time exceeds the derived upper bound.

read the original abstract

We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the $2$-Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper establishes general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated exponentially ergodic Markovian Feller semigroup. It proves improved moment estimates for the solutions and the e-property of the semigroup. Furthermore, it provides quantitative upper bounds for the 2-Wasserstein ε-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as stochastic semilinear equations such as the 1D stochastic Burgers equation.

Significance. If the results hold, the work supplies a unified set of structural hypotheses (local monotonicity plus coercivity and growth) under which Krylov-Bogoliubov plus controllability arguments close in the variational setting, yielding both existence/uniqueness of the invariant measure and quantitative Wasserstein mixing rates. The explicit verification that the listed examples (2D NSE, power-law fluids, Burgers) satisfy the hypotheses in the indicated Gelfand triples is a concrete strength, as is the handling of degenerate additive noise without hidden non-degeneracy assumptions.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'general quantitative conditions' is used without a one-sentence preview of the precise form of the local-monotonicity, coercivity, and growth assumptions; adding such a sentence would improve readability for readers outside the immediate subfield.
  2. The manuscript states that the examples meet the structural hypotheses, but the precise function-space settings (e.g., the Gelfand triple for the power-law fluids on unbounded domains) should be cross-referenced explicitly to the verification statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so there are no specific points requiring point-by-point responses or revisions at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from structural hypotheses

full rationale

The paper derives existence of a unique invariant measure and exponential ergodicity for the Markov semigroup directly from the local monotonicity, coercivity, and growth conditions on the drift together with the variational formulation and degenerate additive noise. These assumptions are used to close a priori estimates that feed into the Krylov-Bogoliubov theorem and yield the e-property plus Wasserstein mixing bounds; the examples are verified to satisfy the same hypotheses in the indicated spaces. No step reduces a derived quantity to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is unverified. The central claims remain independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results in the variational theory of stochastic evolution equations (existence of mild solutions, tightness criteria, etc.) that are treated as known from the literature; no free parameters or new postulated entities are mentioned in the abstract.

axioms (1)
  • standard math Standard variational setting (Gelfand triple, coercivity/growth conditions compatible with local monotonicity) for stochastic evolution equations
    Invoked implicitly to formulate the equations and close the a-priori estimates.

pith-pipeline@v0.9.0 · 5638 in / 1363 out tokens · 38970 ms · 2026-05-23T08:36:16.172343+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

98 extracted references · 98 canonical work pages

  1. [1]

    R. A. Adams and J. J. F. Fournier. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, second edition, 200 3

    work page

  2. [2]

    Agresti and M

    A. Agresti and M. Veraar. Nonlinear parabolic stochasti c evolution equations in critical spaces part I. Stochastic maximal regularity and local existence. Nonlinearity, 35(8):4100–4210, 2022

    work page 2022

  3. [3]

    Agresti and M

    A. Agresti and M. Veraar. Nonlinear parabolic stochasti c evolution equations in critical spaces part II: Blow-up criteria and instataneous regularization. J. Evol. Equ. , 22(2):Paper No. 56, 96, 2022

    work page 2022

  4. [4]

    Agresti and M

    A. Agresti and M. Veraar. The critical variational setti ng for stochastic evolution equations. Probab. Theory Related Fields , 188(3-4):957–1015, 2024

    work page 2024

  5. [5]

    Albeverio, Z

    S. Albeverio, Z. Brze´ zniak, and J.-L. Wu. Existence of g lobal solutions and invariant measures for stochastic differential equations driven by Poisson typ e noise with non-Lipschitz coefficients. J. Math. Anal. Appl. , 371(1):309–322, 2010

    work page 2010

  6. [6]

    Albeverio, A

    S. Albeverio, A. Debussche, and L. Xu. Exponential mixin g of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises. Appl. Math. Optim. , 66(2):273–308, 2012

    work page 2012

  7. [7]

    Barbu and G

    V. Barbu and G. Da Prato. Ergodicity for nonlinear stocha stic equations in variational formulation. Appl. Math. Optim. , 53(2):121–139, 2006

    work page 2006

  8. [8]

    Barbu and G

    V. Barbu and G. Da Prato. The Kolmogorov equation for a 2D- Navier-Stokes stochastic flow in a channel. Nonlinear Anal. , 69(3):940–949, 2008

    work page 2008

  9. [9]

    Barbu and G

    V. Barbu and G. Da Prato. Invariant measures and the Kolmo gorov equation for the stochastic fast diffusion equation. Stochastic Process. Appl. , 120(7):1247–1266, 2010

    work page 2010

  10. [10]

    Bedrossian, M

    J. Bedrossian, M. Coti Zelati, and N. Glatt-Holtz. Inva riant measures for passive scalars in the small noise inviscid limit. Comm. Math. Phys. , 348(1):101–127, 2016

    work page 2016

  11. [11]

    Bedrossian, M

    J. Bedrossian, M. C. Zelati, S. Punshon-Smith, and F. We ber. Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations. Arch. Ration. Mech. Anal. , 237(1):103–145, 2020

    work page 2020

  12. [12]

    Boursier, D

    J. Boursier, D. Chafa ¨ ı, and C. Labb´ e. Universal cutoff for Dyson Ornstein Uhlenbeck process. Probab. Theory Related Fields , 185(1-2):449–512, 2023

    work page 2023

  13. [13]

    Bricmont, A

    J. Bricmont, A. Kupiainen, and R. Lefevere. Exponentia l mixing of the 2D stochastic Navier-Stokes dynamics. Comm. Math. Phys. , 230(1):87–132, 2002

    work page 2002

  14. [14]

    Brze´ zniak, E

    Z. Brze´ zniak, E. Hausenblas, and J. Zhu. 2D stochastic Navier-Stokes equations driven by jump noise. Nonlinear Anal. , 79:122–139, 2013

    work page 2013

  15. [15]

    Brze´ zniak, T

    Z. Brze´ zniak, T. Komorowski, and S. Peszat. Ergodicit y for stochastic equations of Navier-Stokes type. Electron. Commun. Probab. , 27:Paper No. 4, 10, 2022

    work page 2022

  16. [16]

    Brze´ zniak, W

    Z. Brze´ zniak, W. Liu, and J. Zhu. Strong solutions for S PDE with locally monotone coefficients driven by L´ evy noise.Nonlinear Anal. Real World Appl. , 17:283–310, 2014

    work page 2014

  17. [17]

    Brze´ zniak, X

    Z. Brze´ zniak, X. Peng, and J. Zhai. Well-posedness and large deviations for 2D stochastic Navier- Stokes equations with jumps. J. Eur. Math. Soc. (JEMS) , 25(8):3093–3176, 2023

    work page 2023

  18. [18]

    Butkovsky, A

    O. Butkovsky, A. Kulik, and M. Scheutzow. Generalized c ouplings and ergodic rates for SPDEs and other Markov models. Ann. Appl. Probab. , 30(1):1–39, 2020. ERGODICITY AND MIXING FOR STOCHASTIC EVOLUTION EQUATIONS 4 1

    work page 2020

  19. [19]

    L. Chen, C. Ouyang, S. Tindel, and P. Xia. On ergodic prop erties of stochastic PDEs. Stoch. PDE: Anal. Comp. , pages 1–45, 2025. https://doi.org/10.1007/s40072-025-00365-0

    work page doi:10.1007/s40072-025-00365-0 2025

  20. [20]

    Chojnowska-Michalik and B

    A. Chojnowska-Michalik and B. Goldys. Existence, uniq ueness and invariant measures for stochastic semilinear equations on Hilbert spaces. Probab. Theory Related Fields , 102(3):331–356, 1995

    work page 1995

  21. [21]

    Cooperman and K

    W. Cooperman and K. Rowan. Exponential scalar mixing fo r the 2D Navier-Stokes equations with degenerate stochastic forcing, 2024. https://arxiv.org/abs/2408.02459

    work page arXiv 2024

  22. [22]

    Da Prato

    G. Da Prato. Kolmogorov equations for stochastic PDEs . Advanced Courses in Mathematics. CRM Barcelona. Birkh¨ auser Verlag, Basel, 2004

    work page 2004

  23. [23]

    Da Prato, D

    G. Da Prato, D. G¸ atarek, and J. Zabczyk. Invariant meas ures for semilinear stochastic equations. Stochastic Anal. Appl. , 10(4):387–408, 1992

    work page 1992

  24. [24]

    Da Prato and J

    G. Da Prato and J. Zabczyk. Ergodicity for infinite-dimensional systems , volume 229 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 1996

    work page 1996

  25. [25]

    Da Prato and J

    G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions , volume 152 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, second edition, 2 014

    work page

  26. [26]

    Diaconis

    P. Diaconis. The cut-off phenomenon in finite markov chai ns. Proc. Nat. Acad. Sci. U.S.A. , 93(4):1659–1664, 1996

    work page 1996

  27. [27]

    Dong and Y

    Z. Dong and Y. Xie. Ergodicity of stochastic 2D Navier-S tokes equation with L´ evy noise.J. Differ- ential Equations , 251(1):196–222, 2011

    work page 2011

  28. [28]

    W. E, J. C. Mattingly, and Y. Sinai. Gibbsian dynamics an d ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys. , 224(1):83–106, 2001. Dedicated to Joel L. Lebowitz

    work page 2001

  29. [29]

    Ekeland and R

    I. Ekeland and R. T´ emam. Convex analysis and variational problems , volume 28 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Ph iladelphia, PA, english edition, 1999. Translated from the French

    work page 1999

  30. [30]

    Es-Sarhir

    A. Es-Sarhir. Existence and uniqueness of invariant me asures for a class of transition semigroups on Hilbert spaces. J. Math. Anal. Appl. , 353(2):497–507, 2009

    work page 2009

  31. [31]

    Es-Sarhir, M

    A. Es-Sarhir, M. Scheutzow, J. M. T¨ olle, and O. van Gaan s. Invariant measures for monotone SPDEs with multiplicative noise term. Appl. Math. Optim. , 68(2):275–287, 2013

    work page 2013

  32. [32]

    Es-Sarhir and W

    A. Es-Sarhir and W. Stannat. Invariant measures for sem ilinear SPDE’s with local Lipschitz drift coefficients and applications. J. Evol. Equ. , 8(1):129–154, 2008

    work page 2008

  33. [33]

    Es-Sarhir and M.-K

    A. Es-Sarhir and M.-K. von Renesse. Ergodicity of stoch astic curve shortening flow in the plane. SIAM J. Math. Anal. , 44(1):224–244, 2012

    work page 2012

  34. [34]

    Es-Sarhir, M.-K

    A. Es-Sarhir, M.-K. von Renesse, and W. Stannat. Estima tes for the ergodic measure and polynomial stability of plane stochastic curve shortening flow. NoDEA Nonlinear Differential Equations Appl. , 19(6):663–675, 2012

    work page 2012

  35. [35]

    Ferrario and M

    B. Ferrario and M. Zanella. Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise. Discrete Contin. Dyn. Syst. , 44(1):228–262, 2024

    work page 2024

  36. [36]

    Ferrario and M

    B. Ferrario and M. Zanella. Long time behavior of the sto chastic 2D Navier-Stokes equations. Preprint, pages 1–18, 2025. https://arxiv.org/abs/2501.01772

    work page arXiv 2025

  37. [37]

    Flandoli and A

    F. Flandoli and A. Mahalov. Stochastic three-dimensio nal rotating Navier-Stokes equations: aver- aging, convergence and regularity. Arch. Ration. Mech. Anal. , 205(1):195–237, 2012

    work page 2012

  38. [38]

    Flandoli and B

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

    work page 1995

  39. [39]

    Friesen, P

    M. Friesen, P. Jin, J. Kremer, and B. R¨ udiger. Exponent ial ergodicity for stochastic equations of nonnegative processes with jumps. ALEA Lat. Am. J. Probab. Math. Stat. , 20(1):593–627, 2023

    work page 2023

  40. [40]

    S. Geiss. Sharp convex generalizations of stochastic G ronwall inequalities. J. Differential Equations , 392:74–127, 2024

    work page 2024

  41. [41]

    B. Gess, W. Liu, and A. Schenke. Random attractors for lo cally monotone stochastic partial differ- ential equations. J. Differential Equations , 269(4):3414–3455, 2020

    work page 2020

  42. [42]

    Gess and J

    B. Gess and J. M. T¨ olle. Multi-valued, singular stocha stic evolution inclusions. J. Math. Pures Appl. (9), 101(6):789–827, 2014

    work page 2014

  43. [43]

    Gess and J

    B. Gess and J. M. T¨ olle. Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. SIAM J. Math. Anal. , 48(6):4094–4125, 2016. 42 G. BARRERA AND J. M. T ¨OLLE

    work page 2016

  44. [44]

    Glatt-Holtz, J

    N. Glatt-Holtz, J. C. Mattingly, and G. Richards. On uni que ergodicity in nonlinear stochastic partial differential equations. J. Stat. Phys. , 166(3-4):618–649, 2017

    work page 2017

  45. [45]

    Glatt-Holtz, V

    N. Glatt-Holtz, V. ˇSver´ ak, and V. Vicol. On inviscid limits for the stochasticNavier-Stokes equations and related models. Arch. Ration. Mech. Anal. , 217(2):619–649, 2015

    work page 2015

  46. [46]

    M. V. Gnann, J. Hoogendijk, and M. C. Veraar. Higher orde r moments for SPDE with monotone nonlinearities. Stochastics, 96(7):1948–1983, 2024

    work page 1948

  47. [47]

    Goldys and B

    B. Goldys and B. Maslowski. Uniform exponential ergodi city of stochastic dissipative systems. Czechoslovak Math. J. , 51(126)(4):745–762, 2001

    work page 2001

  48. [48]

    Goldys and B

    B. Goldys and B. Maslowski. Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations. J. Funct. Anal. , 226(1):230–255, 2005

    work page 2005

  49. [49]

    Gy¨ ongy and N

    I. Gy¨ ongy and N. V. Krylov. On stochastic equations wit h respect to semimartingales. I. Stochastics, 4(1):1–21, 1980/81

    work page 1980

  50. [50]

    Gy¨ ongy and N

    I. Gy¨ ongy and N. V. Krylov. On stochastics equations wi th respect to semimartingales. II. Itˆ o formula in Banach spaces. Stochastics, 6(3-4):153–173, 1981/82

    work page 1981

  51. [51]

    Gy¨ ongy and D

    I. Gy¨ ongy and D. ˇSiˇ ska. Itˆ o formula for processes taking values in interse ction of finitely many Banach spaces. Stoch. Partial Differ. Equ. Anal. Comput. , 5(3):428–455, 2017

    work page 2017

  52. [52]

    Hairer and J

    M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navie r-Stokes equations with degenerate sto- chastic forcing. Ann. of Math. (2) , 164(3), 2006

    work page 2006

  53. [53]

    Hairer and J

    M. Hairer and J. C. Mattingly. A theory of hypoelliptici ty and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. , 16:no. 23, 658–738, 2011

    work page 2011

  54. [54]

    Hairer and W

    M. Hairer and W. Zhao. Ergodicity of 2D singular stochas tic Navier-Stokes equations. Preprint, pages 1–35, 2024. https://arxiv.org/abs/2411.03482

    work page arXiv 2024

  55. [55]

    Hofmanov´ a, R

    M. Hofmanov´ a, R. Zhu, and X. Zhu. Global existence and n on-uniqueness for 3D Navier-Stokes equations with space-time white noise. Arch. Ration. Mech. Anal. , 247(3):Paper No. 46, 70, 2023

    work page 2023

  56. [56]

    Hofmanov´ a, R

    M. Hofmanov´ a, R. Zhu, and X. Zhu. Non-unique ergodicit y for deterministic and stochas- tic 3D Navier-Stokes and Euler equations. Arch. Rational Mech. Anal. , 249(33):1–54, 2025. https://doi.org/10.1007/s00205-025-02102-2

    work page doi:10.1007/s00205-025-02102-2 2025

  57. [57]

    Komorowski, S

    T. Komorowski, S. Peszat, and T. Szarek. On ergodicity o f some Markov processes. Ann. Probab., 38(4):1401–1443, 2010

    work page 2010

  58. [58]

    N. V. Krylov and B. L. Rozovski ˘ ı. Stochastic evolution equations. In Current problems in mathe- matics, Vol. 14 (Russian) , Itogi Nauki i Tekhniki, pages 71–147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1979

    work page 1979

  59. [59]

    Kuksin, V

    S. Kuksin, V. Nersesyan, and A. Shirikyan. Exponential mixing for a class of dissipative PDEs with bounded degenerate noise. Geom. Funct. Anal. , 30(1):126–187, 2020

    work page 2020

  60. [60]

    Kuksin and A

    S. Kuksin and A. Shirikyan. Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. , 4(2):147–195, 2001

    work page 2001

  61. [61]

    Kuksin and A

    S. Kuksin and A. Shirikyan. Mathematics of two-dimensional turbulence , volume 194 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2012

    work page 2012

  62. [62]

    Kulik and M

    A. Kulik and M. Scheutzow. Generalized couplings and co nvergence of transition probabilities. Probab. Theory Related Fields , 171(1-2):333–376, 2018

    work page 2018

  63. [63]

    Kumar and M

    A. Kumar and M. T. Mohan. Well-posedness of a class of sto chastic partial differential equations with fully monotone coefficients perturbed by L´ evy noise. Anal. Math. Phys. , 14(44), 2024

    work page 2024

  64. [64]

    in the large

    O. A. Ladyˇ zenskaja. Solution “in the large” of the nons tationary boundary value problem for the Navier-Stokes system with two space variables. Comm. Pure Appl. Math. , 12:427–433, 1959

    work page 1959

  65. [65]

    Lasota and T

    A. Lasota and T. Szarek. Lower bound technique in the the ory of a stochastic differential equation. J. Differential Equations , 231(2):513–533, 2006

    work page 2006

  66. [66]

    D. A. Levin, Y. Peres, and E. L. Wilmer. Markov chains and mixing times . American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Pro pp and David B. Wilson

    work page 2009

  67. [67]

    W. Liu. Harnack inequality and applications for stocha stic evolution equations with monotone drifts. J. Evol. Equ. , 9(4):747–770, 2009

    work page 2009

  68. [68]

    W. Liu. Ergodicity of transition semigroups for stocha stic fast diffusion equations. Front. Math. China, 6(3):449–472, 2011. ERGODICITY AND MIXING FOR STOCHASTIC EVOLUTION EQUATIONS 4 3

    work page 2011

  69. [69]

    Liu and M

    W. Liu and M. R¨ ockner. SPDE in Hilbert space with locall y monotone coefficients. J. Funct. Anal. , 259(11):2902–2922, 2010

    work page 2010

  70. [70]

    Liu and M

    W. Liu and M. R¨ ockner. Stochastic partial differential equations: an introduction . Universitext. Springer, Cham, 2015

    work page 2015

  71. [71]

    Liu and J

    W. Liu and J. M. T¨ olle. Existence and uniqueness of inva riant measures for stochastic evolution equations with weakly dissipative drifts. Electron. Commun. Probab. , 16:447–457, 2011

    work page 2011

  72. [72]

    Liu and Z.-y

    Y. Liu and Z.-y. Liu. Relation between the eventual cont inuity and the E-property for Markov-Feller semigroups. Acta Math. Appl. Sin. Engl. Ser. , 40(1):1–16, 2024

    work page 2024

  73. [73]

    M´ alek, J

    J. M´ alek, J. Neˇ cas, M. Rokyta, and M. R˚ uˇ ziˇ cka.Weak and measure-valued solutions to evolution- ary PDEs , volume 13 of Applied Mathematics and Mathematical Computation . Chapman & Hall, London, 1996

    work page 1996

  74. [74]

    Maslowski

    B. Maslowski. Uniqueness and stability of invariant me asures for stochastic differential equations in Hilbert spaces. Stochastics Stochastics Rep. , 28(2):85–114, 1989

    work page 1989

  75. [75]

    Maslowski and J

    B. Maslowski and J. Seidler. On sequentially weakly Fel ler solutions to SPDE’s. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. , 10(2):69–78, 1999

    work page 1999

  76. [76]

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

    work page 2002

  77. [77]

    M´ etivier

    M. M´ etivier. Semimartingales, volume 2 of de Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin-New York, 1982. A course on stochastic process es

    work page 1982

  78. [78]

    M. T. Mohan, K. Sakthivel, and S. S. Sritharan. Ergodici ty for the 3D stochastic Navier-Stokes equations perturbed by L´ evy noise. Math. Nachr. , 292(5):1056–1088, 2019

    work page 2019

  79. [79]

    M. T. Mohan and S. S. Sritharan. Ergodic control of stoch astic Navier-Stokes equation with L´ evy noise. Commun. Stoch. Anal. , 10(3):Article 7, 389–404, 2016

    work page 2016

  80. [80]

    ˇSiˇ ska

    Neelima and D. ˇSiˇ ska. Coercivity condition for higher moment a priori est imates for nonlinear SPDEs and existence of a solution under local monotonicity. Stochastics, 92(5):684–715, 2020

    work page 2020

Showing first 80 references.