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 & 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.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[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.  doi: 10.1081/PDE-100002246.  Google Scholar

[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.  doi: 10.1016/0375-9601(78)90508-X.  Google Scholar

[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.   Google Scholar

[5]

H. H. Chang, P. Oh, D. E. Ingber and S. Huang, Multistable and multistep dynamics in neutrophil differentiation,, MBC Cell Biology, 7 (2006).   Google Scholar

[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.  doi: 10.1007/s00205-011-0471-6.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

M. Erbar, Low noise limit for the invariant measure of a multi-dimensional stochastic Allen-Cahn equation,, \arXiv{1012.2718}., ().   Google Scholar

[9]

T. Funaki, Singular limit for stochastic reaction-diffusion equation nd generation of random interface,, Acta. Math. Sin. (Engl. Ser.), 15 (1999), 407.  doi: 10.1007/BF02650735.  Google Scholar

[10]

T. Funaki, The scaling limit for a stochastic PDE and the separation of phases,, Probab. Theory Ralated Fields, 102 (1995), 221.  doi: 10.1007/BF01213390.  Google Scholar

[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}., ().   Google Scholar

[12]

A. Friedman, "Generalized Functions and Partial Differential Equations,", Prentice-Hall, (1963).   Google Scholar

[13]

C. W. Gardiner, "Handbooks of Stochastic Methods in Physics, Chemistry, and Nautral Sciences,", Springer-Verlag, (1983).   Google Scholar

[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.  doi: 10.4171/IFB/154.  Google Scholar

[15]

S. Kogan, "Electronic Noise and Fluctuations in Solids,", Cambridge University Press, (1996).  doi: 10.1017/CBO9780511551666.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/S0764-4442(98)80067-0.  Google Scholar

[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.   Google Scholar

[19]

J. M. Porrá and J. Masoliver, Bistability driven by white shot noise,, Phys. Rev. E, 47 (1993), 1633.  doi: 10.1103/PhysRevE.47.1633.  Google Scholar

[20]

J. M. Porrá, J. Masoliver and K. Lindenberg, Bistability driven by dichotomous noise,, Phys. Rev., 44 (1991), 4866.  doi: 10.1103/PhysRevA.44.4866.  Google Scholar

[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.  doi: 10.1016/j.crma.2004.12.025.  Google Scholar

[22]

D. Ryter, Conditions for Gibbs-type solutions of Stationary Fokker-Planck equations,, J. Phys. A, 18 (1985), 1111.  doi: 10.1088/0305-4470/18/7/019.  Google Scholar

[23]

L. Schimansky-Geier and C. Zülick, Harmonic noise: effect on bistable systems,, Z. Phys. B-Condensed Matter, 79 (1990), 451.  doi: 10.1007/BF01437657.  Google Scholar

[24]

L. Schimansky-Geier, J. J. Hesse and C. Zülick, Harmonic noise driven bistable dynamics,, Berichte der Bunsengesellschaft für physikalischei Chemie, 95 (1991), 349.  doi: 10.1002/bbpc.19910950321.  Google Scholar

[25]

Z. Schuss, "Theory and Applications of Stochastic Processes, An Analytical Approach,", Springer, (2010).  doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[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.   Google Scholar

[27]

H. Weber, On the short time asymptotic of stochatic Allen-Cahn equation,, Ann. Inst. H. Poincar\`e Probab. Stat., 46 (2010), 965.  doi: 10.1214/09-AIHP333.  Google Scholar

[28]

H. Weber, Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 63 (2010), 1071.  doi: 10.1002/cpa.20323.  Google Scholar

[29]

W. Weidlich and H. Grabert, Renormalied transport equations for the bistable potential model,, Z. Physik B, 36 (1980), 283.  doi: 10.1007/BF01325292.  Google Scholar

[30]

N. Yip, Stochastic motion by mean curvature,, Arch. Rational. Mech. Anal., 144 (1998), 313.  doi: 10.1007/s002050050120.  Google Scholar

[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.   Google Scholar

[32]

L. Zhang, L. Cao and D. Wu, Effect of correlated noises in an optical bistable system,, Physical Review A, 77 (2008).  doi: 10.1103/PhysRevA.77.015801.  Google Scholar

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.  doi: 10.1016/0001-6160(79)90196-2.  Google Scholar

[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.  doi: 10.1081/PDE-100002246.  Google Scholar

[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.  doi: 10.1016/0375-9601(78)90508-X.  Google Scholar

[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.   Google Scholar

[5]

H. H. Chang, P. Oh, D. E. Ingber and S. Huang, Multistable and multistep dynamics in neutrophil differentiation,, MBC Cell Biology, 7 (2006).   Google Scholar

[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.  doi: 10.1007/s00205-011-0471-6.  Google Scholar

[7]

G. Da Prato and J. Zabczyk, "Ergodicity for Infinite-Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[8]

M. Erbar, Low noise limit for the invariant measure of a multi-dimensional stochastic Allen-Cahn equation,, \arXiv{1012.2718}., ().   Google Scholar

[9]

T. Funaki, Singular limit for stochastic reaction-diffusion equation nd generation of random interface,, Acta. Math. Sin. (Engl. Ser.), 15 (1999), 407.  doi: 10.1007/BF02650735.  Google Scholar

[10]

T. Funaki, The scaling limit for a stochastic PDE and the separation of phases,, Probab. Theory Ralated Fields, 102 (1995), 221.  doi: 10.1007/BF01213390.  Google Scholar

[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}., ().   Google Scholar

[12]

A. Friedman, "Generalized Functions and Partial Differential Equations,", Prentice-Hall, (1963).   Google Scholar

[13]

C. W. Gardiner, "Handbooks of Stochastic Methods in Physics, Chemistry, and Nautral Sciences,", Springer-Verlag, (1983).   Google Scholar

[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.  doi: 10.4171/IFB/154.  Google Scholar

[15]

S. Kogan, "Electronic Noise and Fluctuations in Solids,", Cambridge University Press, (1996).  doi: 10.1017/CBO9780511551666.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/S0764-4442(98)80067-0.  Google Scholar

[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.   Google Scholar

[19]

J. M. Porrá and J. Masoliver, Bistability driven by white shot noise,, Phys. Rev. E, 47 (1993), 1633.  doi: 10.1103/PhysRevE.47.1633.  Google Scholar

[20]

J. M. Porrá, J. Masoliver and K. Lindenberg, Bistability driven by dichotomous noise,, Phys. Rev., 44 (1991), 4866.  doi: 10.1103/PhysRevA.44.4866.  Google Scholar

[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.  doi: 10.1016/j.crma.2004.12.025.  Google Scholar

[22]

D. Ryter, Conditions for Gibbs-type solutions of Stationary Fokker-Planck equations,, J. Phys. A, 18 (1985), 1111.  doi: 10.1088/0305-4470/18/7/019.  Google Scholar

[23]

L. Schimansky-Geier and C. Zülick, Harmonic noise: effect on bistable systems,, Z. Phys. B-Condensed Matter, 79 (1990), 451.  doi: 10.1007/BF01437657.  Google Scholar

[24]

L. Schimansky-Geier, J. J. Hesse and C. Zülick, Harmonic noise driven bistable dynamics,, Berichte der Bunsengesellschaft für physikalischei Chemie, 95 (1991), 349.  doi: 10.1002/bbpc.19910950321.  Google Scholar

[25]

Z. Schuss, "Theory and Applications of Stochastic Processes, An Analytical Approach,", Springer, (2010).  doi: 10.1007/978-1-4419-1605-1.  Google Scholar

[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.   Google Scholar

[27]

H. Weber, On the short time asymptotic of stochatic Allen-Cahn equation,, Ann. Inst. H. Poincar\`e Probab. Stat., 46 (2010), 965.  doi: 10.1214/09-AIHP333.  Google Scholar

[28]

H. Weber, Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 63 (2010), 1071.  doi: 10.1002/cpa.20323.  Google Scholar

[29]

W. Weidlich and H. Grabert, Renormalied transport equations for the bistable potential model,, Z. Physik B, 36 (1980), 283.  doi: 10.1007/BF01325292.  Google Scholar

[30]

N. Yip, Stochastic motion by mean curvature,, Arch. Rational. Mech. Anal., 144 (1998), 313.  doi: 10.1007/s002050050120.  Google Scholar

[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.   Google Scholar

[32]

L. Zhang, L. Cao and D. Wu, Effect of correlated noises in an optical bistable system,, Physical Review A, 77 (2008).  doi: 10.1103/PhysRevA.77.015801.  Google Scholar

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