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Global stability in the 2D Ricker equation revisited
Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise
College of Science, National University of Defense Technology, Changsha 410073, China |
The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by $α$-stable noise on the Torus $\mathbb{T}$. By the maximal inequality for stochastic $α$-stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.
References:
[1] |
S. Albeverio, J. Wu and T. Zhang,
Parabolic SPDEs driven by Poissson white noise, Stoch. Proc. Appl., 74 (1998), 21-36.
doi: 10.1016/S0304-4149(97)00112-9. |
[2] |
D. Applebaum, Lévy Processes and Stochastic Calculus in Combridge Studies in Advance Mathematics, Cambridge University Press, 2004. Google Scholar |
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C. Doering, J. Gibbon and C. Levermore,
Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
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D. Dai, Z. Li and Z. Liu,
Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation, Physics Letters A, 372 (2008), 3010-3014.
doi: 10.1016/j.physleta.2008.01.015. |
[5] |
Z. Dai and M. Jiang,
Exponential attractors of the Ginzburg-Landau-BBM equations, J. Math. Res. Expo., 21 (2001), 317-322.
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[6] |
Z. Dong, L. Xu and X. Zhang,
Exponential ergodicity of stochastic Burgers equations, J. Stat. Phys., 154 (2014), 929-949.
doi: 10.1007/s10955-013-0881-y. |
[7] |
Z. Dong, L. Xu and X. Zhang,
Invariance measures of stochastic 2D Navier-Stokes equations driven by α-stable processes, Electron. Comm. Probab., 16 (2011), 678-688.
doi: 10.1214/ECP.v16-1664. |
[8] |
Z. Dong and Y. Xie,
Ergodicity of stochastic 2D Navier-stokes equations with Lévy noise, J. Diff. Eqns., 251 (2011), 196-222.
doi: 10.1016/j.jde.2011.03.015. |
[9] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅰ. Compactness method, Physica D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[10] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅱ. Contraction method, Commun. Math. Phys., 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[11] |
B. Guo, H. Huang and M. Jiang, Ginzberg-Landau Equation, Science Press, Beijing, 2002. Google Scholar |
[12] |
M. Hairer, Ergodicity theory for stochastic PDEs, 2008. Available from: http://www.haier.org/notes/Imperial.pdf. Google Scholar |
[13] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm, Pure Appl. Math., 41 (1998), 891-907.
doi: 10.1002/cpa.3160410704. |
[14] |
C. Kening, G. Ponce and L. Vega,
Well-poesdness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
doi: 10.2307/2939277. |
[15] |
K. Porsezian, R. Murali and A. Malomed,
Modulational instability in linearly coupled complex cubie-quintic Ginzberg-Landau equations, Chaos, Soliton and Fractals, 40 (2009), 1907-1913.
|
[16] |
S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2007. Google Scholar |
[17] |
E. Priola and J. Zabcyzk,
Structrual properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theorey Relat. Fields, 149 (2011), 97-137.
doi: 10.1007/s00440-009-0243-5. |
[18] |
E. Priola, L. Xu and J. Zabczyk,
Exponential mixing for some SPDEs with L$\acute{e}$vy noise, Stoch.& Dynam., 11 (2011), 521-534.
doi: 10.1142/S0219493711003425. |
[19] |
E. Priola, A. Shirkyan, L. Xu and J. Zabczyk,
Exponential ergodicity and regularity for equations with Lévy noise, Stoch. Proc. Appl., 122 (2012), 106-133.
doi: 10.1016/j.spa.2011.10.003. |
[20] |
H. Sakaguchi and B. Malomed,
Stable solitons in coupled Ginzberg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities, PhySica D, 183 (2003), 282-292.
doi: 10.1016/S0167-2789(03)00181-7. |
[21] |
T. Shen and J. Huang,
Well-posedness and dynamics of stochastic fractional model for nonlinear optical fibre materials, Nonlin. Anal.-TMA, 110 (2014), 33-46.
doi: 10.1016/j.na.2014.06.018. |
[22] |
E. Stein, Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, 1970. Google Scholar |
[23] |
X. Sun and Y. Xie,
Ergodicity of stochastic dissipative equations driven by α-stable process, Stoch. Anal. Appl., 32 (2014), 61-76.
doi: 10.1080/07362994.2013.843141. |
[24] |
F. Wang,
Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478.
doi: 10.1016/j.spa.2010.12.002. |
[25] |
B. Wang,
The limit behaviour for the Cauchy problem of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 55 (2002), 481-508.
doi: 10.1002/cpa.10024. |
[26] |
L. Xu,
Ergodicity of the stochastic real Ginzburg-Landau equations driven by α-stable noises, Stoch. Proc. Appl., 123 (2013), 3710-3736.
doi: 10.1016/j.spa.2013.05.002. |
[27] |
L. Xu,
Exponential mixing of 2D SDEs forced by degenerate Lévy noise, J. Evol. Equ., 14 (2014), 249-272.
doi: 10.1007/s00028-013-0212-4. |
show all references
References:
[1] |
S. Albeverio, J. Wu and T. Zhang,
Parabolic SPDEs driven by Poissson white noise, Stoch. Proc. Appl., 74 (1998), 21-36.
doi: 10.1016/S0304-4149(97)00112-9. |
[2] |
D. Applebaum, Lévy Processes and Stochastic Calculus in Combridge Studies in Advance Mathematics, Cambridge University Press, 2004. Google Scholar |
[3] |
C. Doering, J. Gibbon and C. Levermore,
Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
[4] |
D. Dai, Z. Li and Z. Liu,
Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation, Physics Letters A, 372 (2008), 3010-3014.
doi: 10.1016/j.physleta.2008.01.015. |
[5] |
Z. Dai and M. Jiang,
Exponential attractors of the Ginzburg-Landau-BBM equations, J. Math. Res. Expo., 21 (2001), 317-322.
|
[6] |
Z. Dong, L. Xu and X. Zhang,
Exponential ergodicity of stochastic Burgers equations, J. Stat. Phys., 154 (2014), 929-949.
doi: 10.1007/s10955-013-0881-y. |
[7] |
Z. Dong, L. Xu and X. Zhang,
Invariance measures of stochastic 2D Navier-Stokes equations driven by α-stable processes, Electron. Comm. Probab., 16 (2011), 678-688.
doi: 10.1214/ECP.v16-1664. |
[8] |
Z. Dong and Y. Xie,
Ergodicity of stochastic 2D Navier-stokes equations with Lévy noise, J. Diff. Eqns., 251 (2011), 196-222.
doi: 10.1016/j.jde.2011.03.015. |
[9] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅰ. Compactness method, Physica D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[10] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation, Ⅱ. Contraction method, Commun. Math. Phys., 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[11] |
B. Guo, H. Huang and M. Jiang, Ginzberg-Landau Equation, Science Press, Beijing, 2002. Google Scholar |
[12] |
M. Hairer, Ergodicity theory for stochastic PDEs, 2008. Available from: http://www.haier.org/notes/Imperial.pdf. Google Scholar |
[13] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm, Pure Appl. Math., 41 (1998), 891-907.
doi: 10.1002/cpa.3160410704. |
[14] |
C. Kening, G. Ponce and L. Vega,
Well-poesdness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.
doi: 10.2307/2939277. |
[15] |
K. Porsezian, R. Murali and A. Malomed,
Modulational instability in linearly coupled complex cubie-quintic Ginzberg-Landau equations, Chaos, Soliton and Fractals, 40 (2009), 1907-1913.
|
[16] |
S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy noise: An Evolution Equation Approach, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2007. Google Scholar |
[17] |
E. Priola and J. Zabcyzk,
Structrual properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theorey Relat. Fields, 149 (2011), 97-137.
doi: 10.1007/s00440-009-0243-5. |
[18] |
E. Priola, L. Xu and J. Zabczyk,
Exponential mixing for some SPDEs with L$\acute{e}$vy noise, Stoch.& Dynam., 11 (2011), 521-534.
doi: 10.1142/S0219493711003425. |
[19] |
E. Priola, A. Shirkyan, L. Xu and J. Zabczyk,
Exponential ergodicity and regularity for equations with Lévy noise, Stoch. Proc. Appl., 122 (2012), 106-133.
doi: 10.1016/j.spa.2011.10.003. |
[20] |
H. Sakaguchi and B. Malomed,
Stable solitons in coupled Ginzberg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities, PhySica D, 183 (2003), 282-292.
doi: 10.1016/S0167-2789(03)00181-7. |
[21] |
T. Shen and J. Huang,
Well-posedness and dynamics of stochastic fractional model for nonlinear optical fibre materials, Nonlin. Anal.-TMA, 110 (2014), 33-46.
doi: 10.1016/j.na.2014.06.018. |
[22] |
E. Stein, Singular Integrals and Differentiablity Properties of Functions, Princeton University Press, 1970. Google Scholar |
[23] |
X. Sun and Y. Xie,
Ergodicity of stochastic dissipative equations driven by α-stable process, Stoch. Anal. Appl., 32 (2014), 61-76.
doi: 10.1080/07362994.2013.843141. |
[24] |
F. Wang,
Gradient estimate for Ornstein-Uhlenbeck jump processes, Stoch. Proc. Appl., 121 (2011), 466-478.
doi: 10.1016/j.spa.2010.12.002. |
[25] |
B. Wang,
The limit behaviour for the Cauchy problem of the complex Ginzburg-Landau equation, Comm. Pure Appl. Math., 55 (2002), 481-508.
doi: 10.1002/cpa.10024. |
[26] |
L. Xu,
Ergodicity of the stochastic real Ginzburg-Landau equations driven by α-stable noises, Stoch. Proc. Appl., 123 (2013), 3710-3736.
doi: 10.1016/j.spa.2013.05.002. |
[27] |
L. Xu,
Exponential mixing of 2D SDEs forced by degenerate Lévy noise, J. Evol. Equ., 14 (2014), 249-272.
doi: 10.1007/s00028-013-0212-4. |
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