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Effects of white noise in multistable dynamics
1. | School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China, China |
2. | Division of Applied Mathematics, Brown University, Providence, RI 02912, United States |
3. | School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006 |
References:
[1] |
S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Mettal., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commu. Partial Differ. Equ., 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[3] |
A. R. Bulsara, W. C. Schieve and R. F. Gragg, Phase transitions induced by white noise in bistable optical systems, Physics Letters A, 168 (1978), 294-296.
doi: 10.1016/0375-9601(78)90508-X. |
[4] |
S. Brassesco, A. De Masi and E. Presutti, Brownian fluctuations of the interface in the $D=1$ Ginzburg-Landau equation with noise, Ann. Inst. H. Poincaré Probab. Statist., 31 (1995), 81-118. |
[5] |
H. H. Chang, P. Oh, D. E. Ingber and S. Huang, Multistable and multistep dynamics in neutrophil differentiation, MBC Cell Biology, 7 (2006), 11. |
[6] |
S. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional on Markov process on a graph, Arch. Rational Mech. Anal., 203 (2012), 969-1008.
doi: 10.1007/s00205-011-0471-6. |
[7] |
G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridye, 1996.
doi: 10.1017/CBO9780511662829. |
[8] |
M. Erbar, Low noise limit for the invariant measure of a multi-dimensional stochastic Allen-Cahn equation,, \arXiv{1012.2718}., ().
|
[9] |
T. Funaki, Singular limit for stochastic reaction-diffusion equation nd generation of random interface, Acta. Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
[10] |
T. Funaki, The scaling limit for a stochastic PDE and the separation of phases, Probab. Theory Ralated Fields, 102 (1995), 221-288.
doi: 10.1007/BF01213390. |
[11] |
I. Fatkullin and E. Vanden-Eijnden, "Coarsening by Diffusion-Annihilation in a Bistable System Driven by Noise,", 2003. Available from: \url{http://www.cims.nyu.edu/~eve2/gl.pdf}., ().
|
[12] |
A. Friedman, "Generalized Functions and Partial Differential Equations," Prentice-Hall, Englewood Cliffs, NJ, 1963. |
[13] |
C. W. Gardiner, "Handbooks of Stochastic Methods in Physics, Chemistry, and Nautral Sciences," Springer-Verlag, Berlin, 1983. |
[14] |
M. A. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem, Interfaces and Free Boundaries, 9 (2007), 1-30.
doi: 10.4171/IFB/154. |
[15] |
S. Kogan, "Electronic Noise and Fluctuations in Solids," Cambridge University Press, 1996.
doi: 10.1017/CBO9780511551666. |
[16] |
D. Liu, Convergence of the spectral method for stochastic Ginzburh-Landau equation driven by space-times white noise, Comm. Math. Sci., 1 (2003), 361-375. |
[17] |
P. L. Lions and P. Souganidis, Fully nonlinear stochastic partial differential equations: Nonsmooth equations and applications, C. R. Acad. Sci. paris Ser. I Math., 326 (1998), 1085-1092.
doi: 10.1016/S0764-4442(98)80067-0. |
[18] |
P. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Math. Contemp., 19 (2000), 1-29. |
[19] |
J. M. Porrá and J. Masoliver, Bistability driven by white shot noise, Phys. Rev. E, 47 (1993), 1633-1641.
doi: 10.1103/PhysRevE.47.1633. |
[20] |
J. M. Porrá, J. Masoliver and K. Lindenberg, Bistability driven by dichotomous noise, Phys. Rev., 44 (1991), 4866-4875.
doi: 10.1103/PhysRevA.44.4866. |
[21] |
M. G. Reznikoff and G. Vanden-Eijnden, Invariant measures of stochastic partial differential equations and conditioned diffusions, C. R. Math. Acda. Sci. Paris, 340 (2005), 305-308.
doi: 10.1016/j.crma.2004.12.025. |
[22] |
D. Ryter, Conditions for Gibbs-type solutions of Stationary Fokker-Planck equations, J. Phys. A, 18 (1985), 1111-1117.
doi: 10.1088/0305-4470/18/7/019. |
[23] |
L. Schimansky-Geier and C. Zülick, Harmonic noise: effect on bistable systems, Z. Phys. B-Condensed Matter, 79 (1990), 451-460.
doi: 10.1007/BF01437657. |
[24] |
L. Schimansky-Geier, J. J. Hesse and C. Zülick, Harmonic noise driven bistable dynamics, Berichte der Bunsengesellschaft für physikalischei Chemie, 95 (1991), 349-352.
doi: 10.1002/bbpc.19910950321. |
[25] |
Z. Schuss, "Theory and Applications of Stochastic Processes, An Analytical Approach," Springer, New York, 2010.
doi: 10.1007/978-1-4419-1605-1. |
[26] |
J. M. R. de Rueda, G. G. Izús and C. H. Borzi, Critical slowing down on the dynamics of a bistable reaction-diffusion system in the neighborhood of its critical point, J. Stats. Phys., 97 (1999), 803-809. |
[27] |
H. Weber, On the short time asymptotic of stochatic Allen-Cahn equation, Ann. Inst. H. Poincar\`e Probab. Stat., 46 (2010), 965-975.
doi: 10.1214/09-AIHP333. |
[28] |
H. Weber, Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 63 (2010), 1071-1109.
doi: 10.1002/cpa.20323. |
[29] |
W. Weidlich and H. Grabert, Renormalied transport equations for the bistable potential model, Z. Physik B, 36 (1980), 283-293.
doi: 10.1007/BF01325292. |
[30] |
N. Yip, Stochastic motion by mean curvature, Arch. Rational. Mech. Anal., 144 (1998), 313-355.
doi: 10.1007/s002050050120. |
[31] |
F. H. Xiao, G. R. Yan and X. W. Zhang, Effect of signal modulating noise in bistable stochastic dynamical systems, Chinese Phys., 12 (2003), 946-950. |
[32] |
L. Zhang, L. Cao and D. Wu, Effect of correlated noises in an optical bistable system, Physical Review A, 77 (2008), [4 pages].
doi: 10.1103/PhysRevA.77.015801. |
show all references
References:
[1] |
S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Mettal., 27 (1979), 1085-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commu. Partial Differ. Equ., 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[3] |
A. R. Bulsara, W. C. Schieve and R. F. Gragg, Phase transitions induced by white noise in bistable optical systems, Physics Letters A, 168 (1978), 294-296.
doi: 10.1016/0375-9601(78)90508-X. |
[4] |
S. Brassesco, A. De Masi and E. Presutti, Brownian fluctuations of the interface in the $D=1$ Ginzburg-Landau equation with noise, Ann. Inst. H. Poincaré Probab. Statist., 31 (1995), 81-118. |
[5] |
H. H. Chang, P. Oh, D. E. Ingber and S. Huang, Multistable and multistep dynamics in neutrophil differentiation, MBC Cell Biology, 7 (2006), 11. |
[6] |
S. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional on Markov process on a graph, Arch. Rational Mech. Anal., 203 (2012), 969-1008.
doi: 10.1007/s00205-011-0471-6. |
[7] |
G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems," London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridye, 1996.
doi: 10.1017/CBO9780511662829. |
[8] |
M. Erbar, Low noise limit for the invariant measure of a multi-dimensional stochastic Allen-Cahn equation,, \arXiv{1012.2718}., ().
|
[9] |
T. Funaki, Singular limit for stochastic reaction-diffusion equation nd generation of random interface, Acta. Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
[10] |
T. Funaki, The scaling limit for a stochastic PDE and the separation of phases, Probab. Theory Ralated Fields, 102 (1995), 221-288.
doi: 10.1007/BF01213390. |
[11] |
I. Fatkullin and E. Vanden-Eijnden, "Coarsening by Diffusion-Annihilation in a Bistable System Driven by Noise,", 2003. Available from: \url{http://www.cims.nyu.edu/~eve2/gl.pdf}., ().
|
[12] |
A. Friedman, "Generalized Functions and Partial Differential Equations," Prentice-Hall, Englewood Cliffs, NJ, 1963. |
[13] |
C. W. Gardiner, "Handbooks of Stochastic Methods in Physics, Chemistry, and Nautral Sciences," Springer-Verlag, Berlin, 1983. |
[14] |
M. A. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem, Interfaces and Free Boundaries, 9 (2007), 1-30.
doi: 10.4171/IFB/154. |
[15] |
S. Kogan, "Electronic Noise and Fluctuations in Solids," Cambridge University Press, 1996.
doi: 10.1017/CBO9780511551666. |
[16] |
D. Liu, Convergence of the spectral method for stochastic Ginzburh-Landau equation driven by space-times white noise, Comm. Math. Sci., 1 (2003), 361-375. |
[17] |
P. L. Lions and P. Souganidis, Fully nonlinear stochastic partial differential equations: Nonsmooth equations and applications, C. R. Acad. Sci. paris Ser. I Math., 326 (1998), 1085-1092.
doi: 10.1016/S0764-4442(98)80067-0. |
[18] |
P. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis, Math. Contemp., 19 (2000), 1-29. |
[19] |
J. M. Porrá and J. Masoliver, Bistability driven by white shot noise, Phys. Rev. E, 47 (1993), 1633-1641.
doi: 10.1103/PhysRevE.47.1633. |
[20] |
J. M. Porrá, J. Masoliver and K. Lindenberg, Bistability driven by dichotomous noise, Phys. Rev., 44 (1991), 4866-4875.
doi: 10.1103/PhysRevA.44.4866. |
[21] |
M. G. Reznikoff and G. Vanden-Eijnden, Invariant measures of stochastic partial differential equations and conditioned diffusions, C. R. Math. Acda. Sci. Paris, 340 (2005), 305-308.
doi: 10.1016/j.crma.2004.12.025. |
[22] |
D. Ryter, Conditions for Gibbs-type solutions of Stationary Fokker-Planck equations, J. Phys. A, 18 (1985), 1111-1117.
doi: 10.1088/0305-4470/18/7/019. |
[23] |
L. Schimansky-Geier and C. Zülick, Harmonic noise: effect on bistable systems, Z. Phys. B-Condensed Matter, 79 (1990), 451-460.
doi: 10.1007/BF01437657. |
[24] |
L. Schimansky-Geier, J. J. Hesse and C. Zülick, Harmonic noise driven bistable dynamics, Berichte der Bunsengesellschaft für physikalischei Chemie, 95 (1991), 349-352.
doi: 10.1002/bbpc.19910950321. |
[25] |
Z. Schuss, "Theory and Applications of Stochastic Processes, An Analytical Approach," Springer, New York, 2010.
doi: 10.1007/978-1-4419-1605-1. |
[26] |
J. M. R. de Rueda, G. G. Izús and C. H. Borzi, Critical slowing down on the dynamics of a bistable reaction-diffusion system in the neighborhood of its critical point, J. Stats. Phys., 97 (1999), 803-809. |
[27] |
H. Weber, On the short time asymptotic of stochatic Allen-Cahn equation, Ann. Inst. H. Poincar\`e Probab. Stat., 46 (2010), 965-975.
doi: 10.1214/09-AIHP333. |
[28] |
H. Weber, Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 63 (2010), 1071-1109.
doi: 10.1002/cpa.20323. |
[29] |
W. Weidlich and H. Grabert, Renormalied transport equations for the bistable potential model, Z. Physik B, 36 (1980), 283-293.
doi: 10.1007/BF01325292. |
[30] |
N. Yip, Stochastic motion by mean curvature, Arch. Rational. Mech. Anal., 144 (1998), 313-355.
doi: 10.1007/s002050050120. |
[31] |
F. H. Xiao, G. R. Yan and X. W. Zhang, Effect of signal modulating noise in bistable stochastic dynamical systems, Chinese Phys., 12 (2003), 946-950. |
[32] |
L. Zhang, L. Cao and D. Wu, Effect of correlated noises in an optical bistable system, Physical Review A, 77 (2008), [4 pages].
doi: 10.1103/PhysRevA.77.015801. |
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