In this paper, we establish a small time large deviation principle (small time asymptotics) for the two-dimensional stochastic Navier–Stokes equations driven by multiplicative noise, which not only involves the study of the small noise, but also the investigation of the effect of the small, but highly nonlinear, unbounded drifts.
Dans cet article, nous établissons un principe de grandes déviations en temps petit pour l’équation de Navier–Stokes bi-dimensionnelle stochastique conduite par un bruit multiplicatif. Celui-ci nécessite non seulement l’étude d’un bruit faible, mais aussi la compréhension des effets de dérives petites mais non bornées et non linéaires.
References
[1] S. Aida and H. Kawabi. Short time asymptotics of a certain infinite dimensional diffusion process. In Stochastic Analysis and Related Topics, VII (Kusadasi, 1998) 77–124. Progr. Probab. 48. Birkhäuser Boston, Boston, MA, 2001. MR1915450 0976.60077[1] S. Aida and H. Kawabi. Short time asymptotics of a certain infinite dimensional diffusion process. In Stochastic Analysis and Related Topics, VII (Kusadasi, 1998) 77–124. Progr. Probab. 48. Birkhäuser Boston, Boston, MA, 2001. MR1915450 0976.60077
[2] S. Aida and T. S. Zhang. On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16 (2002) 67–78. 0993.60026 10.1023/A:1024868720071[2] S. Aida and T. S. Zhang. On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16 (2002) 67–78. 0993.60026 10.1023/A:1024868720071
[3] M. T. Barlow and M. Yor. Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local time. J. Funct. Anal. 49 (1982) 198–229. 0505.60054 10.1016/0022-1236(82)90080-5[3] M. T. Barlow and M. Yor. Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local time. J. Funct. Anal. 49 (1982) 198–229. 0505.60054 10.1016/0022-1236(82)90080-5
[4] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge, 1992. 0761.60052[4] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge, 1992. 0761.60052
[5] B. Davis. On the Lp-norms of stochastic integrals and other martingales. Duke Math. J. 43 1976 697–704. 0349.60061 10.1215/S0012-7094-76-04354-4 euclid.dmj/1077311944[5] B. Davis. On the Lp-norms of stochastic integrals and other martingales. Duke Math. J. 43 1976 697–704. 0349.60061 10.1215/S0012-7094-76-04354-4 euclid.dmj/1077311944
[6] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Jones and Bartlett, Boston, 1993. 0793.60030[6] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Jones and Bartlett, Boston, 1993. 0793.60030
[7] S. Z. Fang and T. S. Zhang. On the small time behavior of Ornstein–Uhlenbeck processes with unbounded linear drifts. Probab. Theory Related Fields 114 (1999) 487–504. 0932.60071 10.1007/s004400050232[7] S. Z. Fang and T. S. Zhang. On the small time behavior of Ornstein–Uhlenbeck processes with unbounded linear drifts. Probab. Theory Related Fields 114 (1999) 487–504. 0932.60071 10.1007/s004400050232
[8] F. Flandoli and D. Gatarek. Martingale and stationary solution for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 (1995) 367–391. 0831.60072 10.1007/BF01192467[8] F. Flandoli and D. Gatarek. Martingale and stationary solution for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 (1995) 367–391. 0831.60072 10.1007/BF01192467
[9] F. Flandoli. Dissipativity and invariant measures for stochastic Navier–Stokes equations. Nonlinear Differential Equations Appl. 1 (1994) 403–423. MR1300150 0820.35108 10.1007/BF01194988[9] F. Flandoli. Dissipativity and invariant measures for stochastic Navier–Stokes equations. Nonlinear Differential Equations Appl. 1 (1994) 403–423. MR1300150 0820.35108 10.1007/BF01194988
[10] M. Gourcy. A large deviation principle for 2D stochastic Navier–Stokes equation. Stochastic Process. Appl. 117 (2007) 904–927. 1117.60027 10.1016/j.spa.2006.11.001[10] M. Gourcy. A large deviation principle for 2D stochastic Navier–Stokes equation. Stochastic Process. Appl. 117 (2007) 904–927. 1117.60027 10.1016/j.spa.2006.11.001
[11] M. Hairer and J. C. Mattingly. Ergodicity of the 2-D Navier–Stokes equation with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006) 993–1032. 1130.37038 10.4007/annals.2006.164.993[11] M. Hairer and J. C. Mattingly. Ergodicity of the 2-D Navier–Stokes equation with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006) 993–1032. 1130.37038 10.4007/annals.2006.164.993
[12] M. Hino and J. Ramirez. Small-time Gaussian behaviour of symmetric diffusion semigroup. Ann. Probab. 31 (2003) 1254–1295. 1085.31008 10.1214/aop/1055425779 euclid.aop/1055425779[12] M. Hino and J. Ramirez. Small-time Gaussian behaviour of symmetric diffusion semigroup. Ann. Probab. 31 (2003) 1254–1295. 1085.31008 10.1214/aop/1055425779 euclid.aop/1055425779
[13] R. Mikulevicius and B. L. Rozovskii. Global L2-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 (2005) 137–176. 1098.60062 10.1214/009117904000000630 euclid.aop/1108141723[13] R. Mikulevicius and B. L. Rozovskii. Global L2-solutions of stochastic Navier–Stokes equations. Ann. Probab. 33 (2005) 137–176. 1098.60062 10.1214/009117904000000630 euclid.aop/1108141723
[14] S. S. Sritharan and P. Sundar. Large deviation for the two dimensional Navier–Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 (2006) 1636–1659. 1117.60064 10.1016/j.spa.2006.04.001[14] S. S. Sritharan and P. Sundar. Large deviation for the two dimensional Navier–Stokes equations with multiplicative noise. Stochastic Process. Appl. 116 (2006) 1636–1659. 1117.60064 10.1016/j.spa.2006.04.001
[16] S. R. S. Varadhan. Diffusion processes in small time intervals. Comm. Pure. Appl. Math. 20 (1967) 659–685. 0278.60051 10.1002/cpa.3160200404[16] S. R. S. Varadhan. Diffusion processes in small time intervals. Comm. Pure. Appl. Math. 20 (1967) 659–685. 0278.60051 10.1002/cpa.3160200404
[17] T. S. Zhang. On the small time asymptotics of diffusion processes on Hilbert spaces. Ann. Probab. 28 (2000) 537–557. 1044.60071 10.1214/aop/1019160252 euclid.aop/1019160252[17] T. S. Zhang. On the small time asymptotics of diffusion processes on Hilbert spaces. Ann. Probab. 28 (2000) 537–557. 1044.60071 10.1214/aop/1019160252 euclid.aop/1019160252