Stochastic Two-Dimensional Hydrodynamical Systems: Wong-Zakai Approximation and Support Theorem

References

• Aida , S. , Kusuoka , S. , and Stroock , D. 1993 . On the support of Wiener functionals . In Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics ( Elworthy , K.D. , and Ikeda N. , Eds.), Pitman Research Notes in Math. Series 284 , Longman , Harlow , UK , pp. 3 – 34 .
   Google Scholar
• Bally , V. , Millet , A. , and Sanz-Solé , M. 1995 . Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations . Annals of Probability 23 : 178 – 222 .
   Web of Science ®Google Scholar
• Barbu , V. , and Da Prato , G. 2007 . Existence and ergodicity for the two-dimensional stochastic magneto-hydrodynamics equations . Appl. Math. Optim. 56 ( 2 ): 145 – 168 .
   Web of Science ®Google Scholar
• Brzeźniak , Z. , Capiński , M. , and Flandoli , F. 1988 . A Convergence result for stochastic partial differential equations . Stochastics 24 : 423 – 445 .
   Google Scholar
• Brzeźniak , Z. , and Flandoli , F. 1995 . Almost sure approximation of Wong-Zakai type for stochastic partial differential equations . Stochastic Process. Appl. 55 : 329 – 358 .
   Web of Science ®Google Scholar
• Capinsky , M. , and Gatarek , D. 1994 . Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension . J. Funct. Anal. 126 : 26 – 35 .
   Web of Science ®Google Scholar
• Cardon-Weber , C. , and Millet , A. 2001 . A support theorem for a generalized Burgers equation . Potential Analysis 15 : 361 – 408 .
   Web of Science ®Google Scholar
• Chueshov , I. , and Vuillermot , P. 2004 . Non-random invariant sets for some systems of parabolic stochastic partial differential equations . Stoch. Anal. Appl. 22 : 1421 – 1486 .
   Web of Science ®Google Scholar
• Chueshov , I. , and Millet , A. 2010 . Stochastic 2D hydrodynamical type systems: Well-posedness and large deviations . Appl. Math. Optim. 61 : 379 – 420 .
   Web of Science ®Google Scholar
• Constantin , P. , and Foias , C. 1988 . Navier–Stokes Equations . U. of Chicago Press , Chicago .
   Google Scholar
• Da Prato , G. , and Zabczyk , J. 1992 . Stochastic Equations in Infinite Dimensions . Cambridge University Press , Cambridge , UK .
   Google Scholar
• Duan , J. , and Millet , A. 2009 . Large deviations for the Boussinesq equations under random influences . Stoch. Proc. and Appl. 119 : 2052 – 2081 .
   Web of Science ®Google Scholar
• Ferrario , B. 1997 . The Bénard problem with random perturbations: Dissipativity and invariant measures . Nonlin. Diff. Equations and Appl. 4 : 101 – 121 .
   Web of Science ®Google Scholar
• Flandoli , F. , and Gatarek , D. 1995 . Martingale and stationary solutions for stochastic Navier–Stokes equations . Probab. Theor. Related Fields 102 : 367 – 391 .
   Web of Science ®Google Scholar
• Grecksch , W. , and Schmalfuss , B. 1996 . Approximation of the stochastic Navier–Stokes equation . Comp. Appl. Math. 15 : 227 – 239 .
   Web of Science ®Google Scholar
• Gyöngy , I. 1988. On the Approximation of Stochastic Partial Differential Equations. Part I, Stochastics 25:59–85; Part II, Stochastics 26:129–164, 1989.
   Google Scholar
• Gyöngy , I. 1989 . The stability of stochastic partial differential equations and applications . Part I, Stochastics and Stochastic Reports 27:129–150; Part II, Stochastics and Stochastic Reports 27 : 189 – 233 .
   Google Scholar
• Gyöngy , I. , and Shmatkov , A. Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations . Appl. Math. Optim. 54 : 315 – 341 .
   Web of Science ®Google Scholar
• Ikeda , N. , and Watanabe , S. 1981 . Stochastic Differential Equations and Diffusion Processes . North Holland , Amsterdam .
   Google Scholar
• Mackevičius , V. 1986 . On the support of the solution of stochastic differential equations . Livetuvos Matematikow Rinkings 36 ( 1 ): 91 – 98 .
   Google Scholar
• Millet , A. , and Sanz-Solé , M. 1994 . The support of the solution to a hyperbolic SPDE . Probability Theory and Related Fields 98 : 361 – 387 .
   Web of Science ®Google Scholar
• Millet , A. , and Sanz-Solé , M. 1994 . A simple proof of the support theorem for diffusion processes . Séminaire de Probabilités XXVIII, Lecture Notes in Mathematics 1583 : 36 – 48 .
   Google Scholar
• Nakayama , T. 2004 . Support theorem for mild solutions of SDE's in Hilbert spaces . J. Math. Sci. Univ. Tokyo 11 : 245 – 311 .
   Google Scholar
• Protter , P. 1985 . Approximations of solutions of stochastic differential equations driven by semi-Martingales . The Annals of Probability 13 ( 3 ): 716 – 743 .
   Web of Science ®Google Scholar
• Stroock , D.W. , and Varadhan , S.R.S. 1972 . On the support of diffusion processes with applications to the strong maximum principle . Proc. of Sixth Berkeley Sym. Math. Stat. Prob. III. Univ. California Press , Berkeley , pp. 333 – 359 .
   Google Scholar
• Tessitore , G. , and Zabczyk , J. 2006 . Wong-Zakai approximations of stochastic evolution equations . J. Evolution Equations 6 : 621 – 655 .
   Web of Science ®Google Scholar
• Triebel , H. 1978 . Interpolation Theory. Functional Spaces and Differential Operators . North Holland , Amsterdam .
   Google Scholar
• Twardowska , K. 1996 . Wong-Zakai approximations for stochastic differential equations . Acta Applic. Math. 43 : 317 – 359 .
   Web of Science ®Google Scholar
• Twardowska , K. 1996 . An approximation theorem of Wong-Zakai type for stochastic Navier–Stokes equations . Rend. Sem. Mat. Univ. Padova 96 : 15 – 36 .
   Google Scholar
• Twardowska , K. 1997 . On support theorems for stochastic nonlinear partial differential equations . In Stochastic Differential and Difference Equations (Györ, 1996) ( Csiszar , I. , and Michaletzky Gy. , Eds.), Progr. Systems Control Theor. 23 . Birkhäuser , Boston , pp. 309 – 317 .
   Google Scholar
• Vishik , M.I. , Komech , A.I. , and Fursikov , A.V. 1979 . Some mathematical problems of statistical hydromechanics . Russ. Math. Surv. 34 ( 5 ): 149 – 234 .
   Google Scholar
• Wong , E. , and Zakai , M. 1969 . Riemann-Stieltjes approximations of stochastic integrals . Z. Wahrscheinlichkeitstheorie u. Verw. Gebiete 12 : 87 – 97 .
   Web of Science ®Google Scholar