# American Institute of Mathematical Sciences

September  2013, 18(7): 1805-1825. doi: 10.3934/dcdsb.2013.18.1805

## 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

Received  January 2013 Revised  April 2013 Published  May 2013

We study the invariant measure of multistable dynamics under the influence of white noise. We show that the invariant measure exists and in the limit of vanishing white noise, the invariant measure approaches a Dirac type measure concentrated at the most stable equilibria if fluctuations are uniform; however, a lesser stable equilibrium may be selected by the fluctuation if its ability to fluctuate is sufficiently smaller than other stable equilibria. Certain related mathematical issues are also addressed.
Citation: Xinfu Chen, Carey Caginalp, Jianghao Hao, Yajing Zhang. Effects of white noise in multistable dynamics. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1805-1825. doi: 10.3934/dcdsb.2013.18.1805
##### 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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##### 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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