Về hành vi giới hạn của các đo lường tĩnh cho các hệ thống tiến hóa ngẫu nhiên với cường độ nhiễu nhỏ

Science China Mathematics - Tập 63 - Trang 1463-1504 - 2019
Lifeng Chen1, Zhao Dong2, Jifa Jiang1, Jianliang Zhai3
1Mathematics and Science College, Shanghai Normal University, Shanghai, China
2Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
3Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, China

Tóm tắt

Hành vi giới hạn của các quá trình tiến hóa ngẫu nhiên với cường độ nhiễu nhỏ ϵ được điều tra trong các phương pháp dựa trên phân phối. Giả sử μϵ là một đo lường tĩnh cho quá trình ngẫu nhiên Xϵ với cường độ ϵ nhỏ và X0 là một dòng bán trên một không gian Ba lan. Giả sử rằng {μϵ: 0 < ϵ ⩽ ϵ0} là chặt chẽ. Sau đó, tất cả các giới hạn của chúng theo nghĩa yếu đều là bất biến theo X0 và các hỗ trợ của chúng nằm trong trung tâm Birkhoff của X0. Các ứng dụng được thực hiện cho nhiều hệ thống tiến hóa ngẫu nhiên khác nhau, bao gồm các phương trình vi phân ngẫu nhiên thông thường, các phương trình vi phân riêng phần ngẫu nhiên, và các phương trình vi phân chức năng ngẫu nhiên được điều khiển bởi chuyển động Brown hoặc các quá trình Lévy.

Từ khóa

#tiến hóa ngẫu nhiên #đo lường tĩnh #nhiễu nhỏ #hệ thống tiến hóa #phương trình vi phân ngẫu nhiên #chuyển động Brown #quá trình Lévy

Tài liệu tham khảo

Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press, 2009 Benaïm M. Recursive algorithms, urn processes and chaining number of chain recurrent sets. Ergodic Theory Dynam Systems, 1998, 18: 53–87 Benaïm M. Dynamics of stochastic approximation algorithms. In: Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol. 1709. Berlin-Heidelberg: Springer, 1999, 1–68 Benaïm M, Hirsch M W. Stochastic approximation algorithms with constant step size whose average is cooperative. Ann Appl Probab, 1999, 9: 216–241 Billingsley P. Convergence of Probability Measures. New York: John Wiley & Sons, 1999 Bogachev V I, Krylov N V, Röckner M. On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm Partial Differential Equations, 2001, 26: 2037–2080 Breźniak Z, Liu W, Zhu J. Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise. Nonlinear Anal Real World Appl, 2014, 17: 283–310 Chen L, Dong Z, Jiang J, et al. Decomposition formula and stationary measures for stochastic Lotka-Volterra system with applications to turbulent convection. J Math Pures Appl (9), 2019, 125: 43–93 Chen X, Jiang J, Niu L. On Lotka-Volterra equations with identical minimal intrinsic growth rate. SIAM J Appl Dyn Syst, 2015, 14: 1558–1599 Conley C C. Isolated Invariant Sets and the Morse Index. Providence: Amer Math Soc, 1978 Cowieson W, Young L-S. SRB measures as zero-noise limits. Ergodic Theory Dynam Systems, 2005, 25: 1115–1138 Da Prato G, Zabczyk J. Ergodicity for Infinite Dimensional Systems. Cambridge: Cambridge University Press, 1996 Dong Z. On the uniqueness of invariant measure of the Burgers equation driven by Lévy processes. J Theoret Probab, 2008, 21: 322–335 Dong Z, Xie Y C. Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise. Sci China Ser A, 2009, 52: 1497–1524 Dong Z, Xie Y C. Ergodicity of stochastic 2D Navier-Stokes equation with Lévy noise. J Differential Equations, 2011, 251: 196–222 Dong Z, Xu T G. One-dimensional stochastic Burgers equation driven by Lévy processes. J Funct Anal, 2007, 243: 631–678 Es-Sarhir A, Scheutzow M, van Gaans O. Invariant measures for stochastic functional differential equations with superlinear drift term. Differential Integral Equations, 2010, 23: 189–200 Freidlin M I, Wentzell A D. Random Perturbations of Dynamical Systems. New York: Springer, 1998 Garroni M G, Menaldi J L. Green Functions for Second Order Parabolic Integro-Differential Problems. Boca Raton: Chapman & Hall, 1992 Haddock J R, Nkashama M N, Wu J. Asymptotic constancy for pseudo monotone dynamical systems on function spaces. J Differential Equations, 1992, 100: 292–311 Hairer M, Mattingly J C, Scheutzow M. Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab Theory Related Fields, 2011, 149: 223–259 Hale J K. Asymptotic Behavior of Dissipative Systems. Providence: Amer Math Soc, 1988 Hirsch M W. Stability and convergence in strongly monotone dynamical systems. J Reine Angew Math, 1988, 383: 1–53 Hirsch M W. Systems of differential equations which are competitive or cooperative, III: Competing species. Nonlinearity, 1988, 1: 51–71 Hirsch M W. Chain transitive sets for smooth strongly monotone dynamical systems. Dyn Contin Discrete Impuls Syst Ser A Math Anal, 1999, 5: 529–543 Hsu S-B. Ordinary Differential Equations with Applications. Singapore: World Scientific, 2006 Huang W, Ji M, Liu Z, et al. Integral identity and measure estimates for stationary Fokker-Planck equations. Ann Probab, 2015, 43: 1712–1730 Huang W, Ji M, Liu Z, et al. Steady states of Fokker-Planck equations, II: Non-existence. J Dynam Differential Equations, 2015, 27: 743–762 Huang W, Ji M, Liu Z, et al. Stochastic stability of measures in gradient systems. Phys D, 2016, 314: 9–17 Huang W, Ji M, Liu Z, et al. Concentration and limit behaviors of stationary measures. Phys D, 2018, 369: 1–17 Hwang C-R. Laplace’s method revisited: Weak convergence of probability measures. Ann Probab, 1980, 8: 1177–1182 Karatzas I, Shreve S. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1991 Khasminskii R Z. Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor Veroyatn Primen, 1960, 5: 196–214 Khasminskii R Z. Stochastic Stability of Differential Equations. New York: Springer, 2012 Li Y, Xie Y C, Zhang X C. Large deviation principle for stochastic heat equation with memory. Discrete Contin Dyn Syst, 2015, 35: 5221–5237 Li Y, Yi Y. Systematic measures of biological networks I: Invariant measures and Entropy. Comm Pure Appl Math, 2016, 69: 1777–1811 Li Y, Yi Y. Systematic measures of biological networks II: Degeneracy, complexity, and robustness. Comm Pure Appl Math, 2016, 69: 1952–1983 Mallet-Paret J, Sell G R. Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J Differential Equations, 1996, 125: 385–440 Mallet-Paret J, Smith H L. The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J Dynam Differential Equations, 1990, 2: 367–421 Mañé R. Ergodic Theory and Differentiable Dynamics. Berlin: Springer-Verlag, 1987 Mao X R. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing Limited, 2008 Menaldi J-L, Sritharan S S. Stochastic 2-D Navier-Stokes equation. Appl Math Optim, 2002, 46: 31–53 Mohammed S E. Stochastic Functional Differential Equations. Melbourne: Pitman Advanced Publishing Program, 1984 Mumford D. The dawning of the age of stochasticity. In: Mathematics: Frontiers and Perspectives. Providence: Amer Math Soc, 2000, 197–218 Poláčik P. Convergence in smooth strongly monotone flows defined by semilinear parabolic equations. J Differential Equations, 1989, 79: 89–110 Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics, 1995 Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer, 2012 van den Driessche P, Zou X F. Global attractivity in delayed Hopfield neural network models. SIAM J Appl Math, 1998, 58: 1878–1890 Young L S. What are SRB measures, and which dynamical systems have them? J Stat Phys, 2002, 108: 733–754 Zhang X C. Exponential ergodicity of non-Lipschitz stochastic differential equations. Proc Amer Math Soc, 2009, 137: 329–337 Zhang X C. On stochastic evolution equations with non-Lipschitz coefficients. Stoch Dyn, 2009, 9: 549–595