Electronic Communications in Probability

Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

Wei Liu and Jonas Toelle

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Abstract

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 40, 447-457.

Dates
Accepted: 22 August 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261996

Digital Object Identifier
doi:10.1214/ECP.v16-1643

Mathematical Reviews number (MathSciNet)
MR2831083

Zentralblatt MATH identifier
1244.60062

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Keywords
stochastic evolution equation invariant measure dissipative $p$-Laplace equation fast diffusion equation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Liu, Wei; Toelle, Jonas. Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts. Electron. Commun. Probab. 16 (2011), paper no. 40, 447--457. doi:10.1214/ECP.v16-1643. https://projecteuclid.org/euclid.ecp/1465261996


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References

  • V. Barbu, G. Da Prato. Ergodicity for nonlinear stochastic equations in variational formulation. Appl. Math. Optim. 53 (2006), no. 2, 121–139.
  • V. Barbu, G. Da Prato. Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation. Stochastic Process. Appl. 120 (2010), no. 7, 1247–1266.
  • V. Barbu, G. Da Prato, M. Röckner. Stochastic porous media equations and self-organized criticality. Comm. Math. Phys. 285 (2009), no. 3, 901–923.
  • J. G. Berryman, C. J. Holland. Stability of the separable solution for fast diffusion. Arch. Rational Mech. Anal. 74 (1980), no. 4, 379–388.
  • V. I. Bogachev, G. Da Prato, M. Röckner. Invariant measures of generalized stochastic equations of porous media. (Russian) Dokl. Akad. Nauk 396 (2004), no. 1, 7–11.
  • I. Ciotir, J. M. Tölle. Convergence of solutions to the stochastic $p$-Laplace equations as $p$ goes to $1$, Preprint, (2010), BiBoS-Preprint 11-01-371
  • G. Da Prato, M. Röckner, B. L. RozovskiÄ­, F.-Y. Wang. Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Comm. Partial Differential Equations 31 (2006), no. 1-3, 277–291.
  • G. Da Prato, J. Zabczyk. 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
  • E. DiBenedetto. Degenerate parabolic equations. Universitext. Springer-Verlag, New York, 1993. xvi+387 pp. ISBN: 0-387-94020-0
  • A. Es-Sarhir, M.-K. von Renesse. Ergodicity of stochastic curve shortening flow in the plane. Preprint, to appear in SIAM J. Math. Anal. (2010), arXiv:1003.2074
  • A. Es-Sarhir, M.-K. von Renesse, W. Stannat. Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow. Preprint (2010), arXiv:1008.1961
  • B. Gess, W. Liu, M. Röckner. Random attractors for a class of stochastic partial differential equations driven by general additive noise. J. Diff. Equations 251 (2011), no. 4–5, 1225–1253.
  • N. V. Krylov, B. L. RozovskiÄ­. Stochastic evolution equations. (Russian) Current problems in mathematics, Vol. 14 (Russian), pp. 71–147, 256, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979.
  • W. Liu. Harnack inequality and applications for stochastic evolution equations with monotone drifts. J. Evol. Equ. 9 (2009), no. 4, 747–770.
  • W. Liu. On the stochastic $p$-Laplace equation. J. Math. Anal. Appl. 360 (2009), no. 2, 737–751.
  • W. Liu. Large deviations for stochastic evolution equations with small multiplicative noise.. Appl. Math. Optim. 61 (2010), no. 1, 27–56.
  • W. Liu, M. Röckner. SPDE in Hilbert space with locally monotone coefficients. J. Funct. Anal. 259 (2010), no. 11, 2902–2922.
  • W. Liu, F.-Y. Wang. Harnack inequality and strong Feller property for stochastic fast-diffusion equations. J. Math. Anal. Appl. 342 (2008), no. 1, 651–662.
  • É. Pardoux. Équations aux dérivées partielles stochastiques de type monotone. (French) Séminaire sur les Équations aux Dérivées Partielles (1974-1975), III, Exp. No. 2, 10 pp. Collège de France, Paris, 1975.
  • C. Prévôt, M. Röckner. A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007. vi+144 pp. ISBN: 978-3-540-70780-6; 3-540-70780-8
  • Ph. Rosenan. Fast and super fast diffusion processes. Phys. Rev. Lett. 74 (1995), no. 11, 7–14.
  • F.-Y. Wang. Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35 (2007), no. 4, 1333–1350.
  • X. Zhang. On stochastic evolution equations with non-Lipschitz coefficients. Stoch. Dyn. 9 (2009), no. 4, 549–595.