ERGODICITY AND WEAK CONVERGENCE OF TRANSITION PROBABILITIES FOR THE 2D PRIMITIVE EQUATIONS WITH MULTIPLICATIVE NOISE

doi:10.1007/s10473-025-0603-y

Abstract

Abstract: This paper investigates the ergodicity and weak convergence of transition proba- bilities for two-dimensional stochastic primitive equations driven by multiplicative noise. The existence of invariant measures is established using the classical Krylov-Bogoliubov theory. The uniqueness of invariant measures and the weak convergence of transition probabilities are demonstrated through the application of the asymptotic coupling method and Foias-Prodi estimate.

Key words: stochastic primitive equations, invariant measure, ergodicity, weak convergence of the transition probabilities, asymptotic coupling method, Foias-Prodi esti-mate.

CLC Number:

35Q35

Jintao LI, Hongjun GAO. ERGODICITY AND WEAK CONVERGENCE OF TRANSITION PROBABILITIES FOR THE 2D PRIMITIVE EQUATIONS WITH MULTIPLICATIVE NOISE[J].Acta mathematica scientia,Series B, 2025, 45(6): 2354-2390.

Trendmd

References

[1] Boulvard P M, Gao P, Nersesyan V. Controllability and ergodicity of three dimensional primitive equations driven by a finite-dimensional force. Arch Ration Mech Anal, 2023, 247(1): Paper 2
[2] Cao C, Titi E S. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann of Math, 2007, 166(1): 245-267
[3] Constantin P, Glatt-Holtz N, Vicol V. Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations. Comm Math Phys, 2014, 330(2): 819-857
[4] Da Prato G, Zabczyk J.Ergodicity for Infinite-Dimensional Systems. London Math Soc Lecture Note Ser, 229. Cambridge: Cambridge University Press, 1996
[5] Da Prato G, Zabczyk J.Stochastic Equations in Infinite Dimensions. Encyclopedia Math Appl, 152. Cambridge: Cambridge University Press, 2014
[6] Debussche A, Glatt-Holtz N, Temam R. Local martingale and pathwise solutions for an abstract fluids model. Phys D, 2011, 240(14/15): 1123-1144
[7] Debussche A, Glatt-Holtz N, Temam R, Ziane M. Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise. Nonlinearity, 2012, 25(7): 2093-2118
[8] Ewald B, Petcu M, Temam R. Stochastic solutions of the two-dimensional primitive equations of the ocean and atmosphere with an additive noise. Anal Appl (Singap), 2007, 5(2): 183-198
[9] Ferrario B, Zanella M. Uniqueness of the invariant measure and asymptotic stability for the 2D Navier-Stokes equations with multiplicative noise. Discrete Contin Dyn Syst, 2024, 44(1): 228-262
[10] Foiaş C, Prodi G. Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2. Rend Sem Mat Univ Padova,1967, 39: 1-34
[11] Gao H, Sun C. Well-posedness and large deviations for the stochastic primitive equations in two space dimensions. Commun Math Sci, 2012, 10(2): 575-593
[12] Glatt-Holtz N, Kukavica I, Vicol V, Ziane M. Existence and regularity of invariant measures for the three dimensional stochastic primitive equations. J Math Phys, 2014, 55(5): 051504
[13] Glatt-Holtz N, Martinez V R, Richards G H.On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation. arXiv: 2103.12942
[14] Glatt-Holtz N, Mattingly J C, Richards G. On unique ergodicity in nonlinear stochastic partial differential equations. J Stat Phys, 2017, 166(3/4): 618-649
[15] Glatt-Holtz N, Temam R. Pathwise solutions of the 2-D stochastic primitive equations. Appl Math Optim, 2011, 63(3): 401-433
[16] Glatt-Holtz N, Ziane M. The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete Contin Dyn Syst Ser B, 2008, 10(4): 801-822
[17] Glatt-Holtz N, Ziane M. Strong pathwise solutions of the stochastic Navier-Stokes system. Adv Differential Equations, 2009, 14(5/6): 567-600
[18] Guo B, Huang D. 3D stochastic primitive equations of the large-scale ocean: global well-posedness and attractors. Comm Math Phys, 2009, 286(2): 697-723
[19] Hairer M, Mattingly J C. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann of Math, 2006, 164(3): 993-1032
[20] Hairer M, Mattingly J C, Scheutzow M. Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations. Probab Theory Related Fields, 2011, 149(1/2): 223-259
[21] Huang D, Shen T, Zheng Y. Ergodicity of two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise. Appl Math Lett, 2020, 102: 106146
[22] Komorowski T, Peszat S, Szarek T. On ergodicity of some Markov processes. Ann Probab, 2010, 38(4): 1401-1443
[23] Kuksin S, Nersesyan V, Shirikyan A. Exponential mixing for a class of dissipative PDEs with bounded degenerate noise. Geom Funct Anal, 2020, 30(1): 126-187
[24] Kuksin S, Shirikyan A.Mathematics of Two-dimensional Turbulence. Cambridge Tracts in Math, 194. Cambridge: Cambridge University Press, 2012
[25] Kulik A, Scheutzow M. Generalized couplings and convergence of transition probabilities. Probab Theory Related Fields, 2018, 171(1/2): 333-376
[26] Lions J L, Temam R, Wang S. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 1992, 5(2): 237-288
[27] Lions J L, Temam R, Wang S. On the equations of the large scale ocean. Nonlinearity, 1992, 5(5): 1007-1053
[28] Lions J L, Temam R, Wang S. Mathematical theory for the coupled atmosphere-ocean models. J Math Pures Appl, 1995, 74(2): 105-163
[29] Odasso C. Exponential mixing for stochastic PDEs: The non-additive case. Probab Theory Related Fields, 2008, 140(1/2): 41-82
[30] Ondreját M. Brownian representations of cylindrical local martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces. Czechoslovak Math J,2005, 55(130): 1003-1039
[31] Sun C, Qiu Z, Tang Y. Ergodicity for two class stochastic partial differential equations with anisotropic viscosity. Statist Probab Lett, 2024, 207: Paper 110022
[32] Zeitlin V.Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models. Oxford: Oxford University Press, 2018