Support Theorem for a Stochastic Cahn-Hilliard Equation

Lijun Bo (Xidian University)
Kehua Shi (Xiamen University)
Yongjin Wang (Nankai University)

Abstract


In this paper, we establish a Stroock-Varadhan support theorem for the global mild solution to a $d$ ($d\leq 3$)-dimensional stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise

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Pages: 484-525

Publication Date: May 1, 2010

DOI: 10.1214/EJP.v15-760

References

  1. Bally, Vlad; Millet, Annie; Sanz-Solé, Marta. Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23 (1995), no. 1, 178--222. MR1330767 (96d:60091)
  2. Blowey, J. F.; Elliott, C. M. The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. European J. Appl. Math. 3 (1992), no. 2, 147--179. MR1166255 (93g:80007)
  3. Bo, Lijun; Shi, Kehua; Wang, Yongjin. Jump type Cahn-Hilliard equations with fractional noises. Chin. Ann. Math. Ser. B 29 (2008), no. 6, 663--678. MR2470622 (2010c:35075)
  4. Bo, Lijun; Wang, Yongjin. Stochastic Cahn-Hilliard partial differential equations with Lévy spacetime white noises. Stoch. Dyn. 6 (2006), no. 2, 229--244. MR2239091 (2007i:60074)
  5. Cahn. J, Hilliard. J. Free energy for a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2 (1958), 258--267.
  6. Cardon-Weber, Caroline. Cahn-Hilliard stochastic equation: existence of the solution and of its density. Bernoulli 7 (2001), no. 5, 777--816. MR1867082 (2002i:60109)
  7. Cardon-Weber, Caroline; Millet, Annie. A support theorem for a generalized Burgers SPDE. Potential Anal. 15 (2001), no. 4, 361--408. MR1856154 (2002g:60097)
  8. Da Prato, Giuseppe; Debussche, Arnaud. Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26 (1996), no. 2, 241--263. MR1359472 (96k:35200)
  9. Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems.London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7 MR1417491 (97k:60165)
  10. Debussche, Arnaud; Dettori, Lucia. On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24 (1995), no. 10, 1491--1514. MR1327930 (96c:35080)
  11. Gyöngy, István. Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process. Appl. 73 (1998), no. 2, 271--299. MR1608641 (99b:60091)
  12. Gyöngy, István; Nualart, David. On the stochastic Burgers' equation in the real line. Ann. Probab. 27 (1999), no. 2, 782--802. MR1698967 (2000f:60091)
  13. Millet, Annie; Sanz-Solé, Marta. The support of the solution to a hyperbolic SPDE. Probab. Theory Related Fields 98 (1994), no. 3, 361--387. MR1262971 (95a:60089)
  14. Nakayama, Toshiyuki. Support theorem for mild solutions of SDE's in Hilbert spaces. J. Math. Sci. Univ. Tokyo 11 (2004), no. 3, 245--311. MR2097527 (2005k:60069)
  15. Novick-Cohen, Amy; Segel, Lee A. Nonlinear aspects of the Cahnmhy Hilliard equation. Phys. D 10 (1984), no. 3, 277--298. MR0763473 (85k:35120)
  16. Nualart, D.; Pardoux, É. White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1992), no. 1, 77--89. MR1172940 (93h:60093)
  17. Stroock, Daniel W.; Varadhan, S. R. S. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 333--359. Univ. California Press, Berkeley, Calif., 1972. MR0400425 (53 #4259)
  18. Temam, Roger. Infinite-dimensional dynamical systems in mechanics and physics.Second edition.Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. xxii+648 pp. ISBN: 0-387-94866-X MR1441312 (98b:58056)
  19. Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV---1984, 265--439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085 (88a:60114)
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