## Bernoulli

• Bernoulli
• Volume 23, Number 2 (2017), 1179-1201.

### Irreducibility of stochastic real Ginzburg–Landau equation driven by $\alpha$-stable noises and applications

Ran Wang, Jie Xiong, and Lihu Xu

#### Abstract

We establish the irreducibility of stochastic real Ginzburg–Landau equation with $\alpha$-stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moderate deviation principle by constructing some Lyapunov test functions.

#### Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 1179-1201.

Dates
Revised: August 2015
First available in Project Euclid: 4 February 2017

https://projecteuclid.org/euclid.bj/1486177396

Digital Object Identifier
doi:10.3150/15-BEJ773

Mathematical Reviews number (MathSciNet)
MR3606763

Zentralblatt MATH identifier
06701623

#### Citation

Wang, Ran; Xiong, Jie; Xu, Lihu. Irreducibility of stochastic real Ginzburg–Landau equation driven by $\alpha$-stable noises and applications. Bernoulli 23 (2017), no. 2, 1179--1201. doi:10.3150/15-BEJ773. https://projecteuclid.org/euclid.bj/1486177396

#### References

• [1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge: Cambridge Univ. Press.
• [2] Brzeźniak, Z., Liu, W. and Zhu, J. (2014). Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise. Nonlinear Anal. Real World Appl. 17 283–310.
• [3] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDEs. Advanced Courses in Mathematics. CRM Barcelona. Basel: Birkhäuser.
• [4] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge: Cambridge Univ. Press.
• [5] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. New York: Springer.
• [6] Dong, Z. (2008). On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes. J. Theoret. Probab. 21 322–335.
• [7] Dong, Z. and Xie, Y. (2011). Ergodicity of stochastic 2D Navier–Stokes equation with Lévy noise. J. Differential Equations 251 196–222.
• [8] Dong, Z., Xu, L. and Zhang, X. (2011). Invariant measures of stochastic 2D Navier–Stokes equations driven by $\alpha$-stable processes. Electron. Commun. Probab. 16 678–688.
• [9] Dong, Z., Xu, T. and Zhang, T. (2009). Invariant measures for stochastic evolution equations of pure jump type. Stochastic Process. Appl. 119 410–427.
• [10] Doob, J.L. (1948). Asymptotic properties of Markoff transition prababilities. Trans. Amer. Math. Soc. 63 393–421.
• [11] Down, D., Meyn, S.P. and Tweedie, R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671–1691.
• [12] Eckmann, J.-P. and Hairer, M. (2001). Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219 523–565.
• [13] Funaki, T. and Xie, B. (2009). A stochastic heat equation with the distributions of Lévy processes as its invariant measures. Stochastic Process. Appl. 119 307–326.
• [14] Masuda, H. (2007). Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. 117 35–56.
• [15] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Encyclopedia of Mathematics and Its Applications 113. Cambridge: Cambridge Univ. Press.
• [16] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Berlin: Springer.
• [17] Priola, E., Shirikyan, A., Xu, L. and Zabczyk, J. (2012). Exponential ergodicity and regularity for equations with Lévy noise. Stochastic Process. Appl. 122 106–133.
• [18] Priola, E., Xu, L. and Zabczyk, J. (2011). Exponential mixing for some SPDEs with Lévy noise. Stoch. Dyn. 11 521–534.
• [19] Priola, E. and Zabczyk, J. (2011). Structural properties of semilinear SPDEs driven by cylindrical stable processes. Probab. Theory Related Fields 149 97–137.
• [20] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
• [21] Wu, L. (2001). Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl. 91 205–238.
• [22] Xu, L. (2013). Ergodicity of the stochastic real Ginzburg–Landau equation driven by $\alpha$-stable noises. Stochastic Process. Appl. 123 3710–3736.
• [23] Xu, L. (2014). Exponential mixing of 2D SDEs forced by degenerate Lévy noises. J. Evol. Equ. 14 249–272.