Abstract
It is shown that stochastic equations can have stable solutions. In particular, there exists stochastic dynamics for which the motion is both ergodic and stable, so that all trajectories merge with time. We discuss this in the context of Monte Carlo-type dynamics, and study the convergence of nearby trajectories as the number of degrees of freedom goes to infinity and as a critical point is approached. A connection with critical slowdown is suggested.
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References
H. Haken, ed.,Synergetics (Springer, Berlin, 1977).
J. P. Crutchfleld, J. D. Farmer, and B. A. Huberman,Phys. Rep. 92:47 (1982) and references therein.
R. Z. Khas'minskii,Theor. Prob. Appl. 12:144 (1967); A. Carverhill,Stochastics 14:209 (1985), and references therein.
V. I. Oseledec,Trans. Moscow Math. Soc. 19:197 (1968).
I. Shimada and T. Nagashima,Prog. Theor. Phys. 61:1605 (1979).
G. Parisi,Nucl. Phys. B180[FS2]:378 (1981); G. Parisi,Nucl. Phys. B205[FS5]:337 (1982).
O. Martin, Nucl. Phys.B251[FS13]:425 (1985).
N. Krasovskii,Stability of Motion (Stanford University Press, Palo Alto, California, 1963), Chap. 5.
G. Benettin, C. Froeschle, and J. P. Scheidecker,Phys. Rev. B19:2454 (1979).
J. D. Farmer,Physica 4D(3):366 (1982); P. Manneville,Macroscopic Modelling of Turbulent Flows and Fluid Mixtures, O. Pironneau, ed. (Springer, Berlin, 1985).
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Martin, O. Lyapunov exponents of stochastic dynamical systems. J Stat Phys 41, 249–261 (1985). https://doi.org/10.1007/BF01020611
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DOI: https://doi.org/10.1007/BF01020611