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Volumn 191, Issue 1, 2003, Pages 1-54

Geometric singular perturbation theory for stochastic differential equations

Author keywords

Concentration of measure; Dynamic bifurcations; First exit times; Invariant manifolds; Singular perturbations; Slow fast systems; Stochastic differential equations

Indexed keywords


EID: 0037950069     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-0396(03)00020-2     Document Type: Article
Times cited : (89)

References (28)
  • 2
    • 0001236639 scopus 로고
    • Petites perturbations aléatoires des systèmes dynamiques: Développements asymptotiques
    • R. Azencott, Petites perturbations aléatoires des systèmes dynamiques: développements asymptotiques, Bull. Sci. Math. (2) 109 (1985) 253-308.
    • (1985) Bull. Sci. Math. , vol.109 , Issue.2 , pp. 253-308
    • Azencott, R.1
  • 5
    • 0036441370 scopus 로고    scopus 로고
    • A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential
    • DOI 10.1214/aoap/1037125869
    • N. Berglund, B. Gentz, A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential, Ann. Appl. Probab. 12 (4) (2002) 1419-1470 DOI 10.1214/aoap/1037125869.
    • (2002) Ann. Appl. Probab. , vol.12 , Issue.4 , pp. 1419-1470
    • Berglund, N.1    Gentz, B.2
  • 6
    • 0041387669 scopus 로고    scopus 로고
    • The effect of additive noise on dynamical hysteresis
    • DOI 10.1088/0951-7715/15/3/305
    • N. Berglund, B. Gentz, The effect of additive noise on dynamical hysteresis, Nonlinearity 15 (3) (2002) 605-632 DOI 10.1088/0951-7715/15/3/305.
    • (2002) Nonlinearity , vol.15 , Issue.3 , pp. 605-632
    • Berglund, N.1    Gentz, B.2
  • 7
    • 0036004914 scopus 로고    scopus 로고
    • Pathwise description of dynamic pitchfork bifurcations with additive noise
    • DOI 10.1007/s004400100174
    • N. Berglund, B. Gentz, Pathwise description of dynamic pitchfork bifurcations with additive noise, Probab. Theory Related Fields 122 (3) (2002) 341-388 DOI 10.1007/s004400100174.
    • (2002) Probab. Theory Related Fields , vol.122 , Issue.3 , pp. 341-388
    • Berglund, N.1    Gentz, B.2
  • 8
    • 0038701464 scopus 로고    scopus 로고
    • Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times
    • preprint WIAS-767
    • A. Bovier, M. Eckhoff, V. Gayrard, M. Klein, Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times, preprint WIAS-767, 2002.
    • (2002)
    • Bovier, A.1    Eckhoff, M.2    Gayrard, V.3    Klein, M.4
  • 9
    • 0038025104 scopus 로고    scopus 로고
    • Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues
    • preprint WIAS-768
    • A. Bovier, V. Gayrard, M. Klein, Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues, preprint WIAS-768, 2002.
    • (2002)
    • Bovier, A.1    Gayrard, V.2    Klein, M.3
  • 10
    • 21844482698 scopus 로고
    • Attractors for random dynamical systems
    • H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (3) (1994) 365-393.
    • (1994) Probab. Theory Related Fields , vol.100 , Issue.3 , pp. 365-393
    • Crauel, H.1    Flandoli, F.2
  • 11
    • 0020500503 scopus 로고
    • On the exponential exit law in the small parameter exit problem
    • M.V. Day, On the exponential exit law in the small parameter exit problem, Stochastics 8 (1983) 297-323.
    • (1983) Stochastics , vol.8 , pp. 297-323
    • Day, M.V.1
  • 12
    • 0038363174 scopus 로고
    • On the exit law from saddle points
    • M.V. Day, On the exit law from saddle points, Stochastic Process. Appl. 60 (1995) 287-311.
    • (1995) Stochastic Process. Appl. , vol.60 , pp. 287-311
    • Day, M.V.1
  • 13
    • 34250627892 scopus 로고
    • Geometric singular perturbation theory for ordinary differential equations
    • N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1) (1979) 53-98.
    • (1979) J. Differential Equations , vol.31 , Issue.1 , pp. 53-98
    • Fenichel, N.1
  • 14
    • 0000030174 scopus 로고
    • Asymptotic series and exit time probabilities
    • W.H. Fleming, M.R. James, Asymptotic series and exit time probabilities, Ann. Probab. 20 (3) (1992) 1369-1384.
    • (1992) Ann. Probab. , vol.20 , Issue.3 , pp. 1369-1384
    • Fleming, W.H.1    James, M.R.2
  • 15
    • 0035532611 scopus 로고    scopus 로고
    • On stable oscillations and equilibriums induced by small noise
    • M.I. Freidlin, On stable oscillations and equilibriums induced by small noise, J. Statist. Phys. 103 (2001) 283-300.
    • (2001) J. Statist. Phys. , vol.103 , pp. 283-300
    • Freidlin, M.I.1
  • 17
    • 0000057019 scopus 로고
    • Application of A. M. Lyapunov's theory of stability to the theory of differential equations with small coefficients in the derivatives
    • I.S. Gradšteǐn, Application of A. M. Lyapunov's theory of stability to the theory of differential equations with small coefficients in the derivatives, Mat. Sbornik N. S. 32 (74) (1953) 263-286.
    • (1953) Mat. Sbornik N. S. , vol.32 , Issue.74 , pp. 263-286
    • Gradšteǐn, I.S.1
  • 18
    • 0018506677 scopus 로고
    • Slowly varying jump and transition phenomena associated with algebraic bifurcation problems
    • R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problems, SIAM J. Appl. Math. 37 (1) (1979) 69-106.
    • (1979) SIAM J. Appl. Math. , vol.37 , Issue.1 , pp. 69-106
    • Haberman, R.1
  • 19
    • 0002316532 scopus 로고
    • Geometric singular perturbation theory
    • (Montecatini Terme, 1994), Springer, Berlin
    • C.K.R.T. Jones, Geometric singular perturbation theory, in: Dynamical systems (Montecatini Terme, 1994), Springer, Berlin, 1995, pp. 44-118.
    • (1995) Dynamical Systems , pp. 44-118
    • Jones, C.K.R.T.1
  • 20
    • 51249184996 scopus 로고
    • The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point
    • Y. Kifer, The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point, Israel J. Math. 40 (1) (1981) 74-96.
    • (1981) Israel J. Math. , vol.40 , Issue.1 , pp. 74-96
    • Kifer, Y.1
  • 21
    • 0016549570 scopus 로고
    • Exchange of stabilities in autonomous systems
    • N.R. Lebovitz, R.J. Schaar, Exchange of stabilities in autonomous systems, Stud. Appl. Math. 54 (3) (1975) 229-260.
    • (1975) Stud. Appl. Math. , vol.54 , Issue.3 , pp. 229-260
    • Lebovitz, N.R.1    Schaar, R.J.2
  • 22
    • 0001915231 scopus 로고
    • Exchange of stabilities in autonomous systems. II. Vertical bifurcation
    • N.R. Lebovitz, R.J. Schaar, Exchange of stabilities in autonomous systems. II. Vertical bifurcation, Stud. Appl. Math. 56 (1) (1976/77) 1-50.
    • (1976) Stud. Appl. Math. , vol.56 , Issue.1 , pp. 1-50
    • Lebovitz, N.R.1    Schaar, R.J.2
  • 24
    • 0000881312 scopus 로고
    • Persistence of stability loss for dynamical bifurcations I
    • A.I. Neǐshtadt, Persistence of stability loss for dynamical bifurcations I, Differential Equations 23 (1987) 1385-1391.
    • (1987) Differential Equations , vol.23 , pp. 1385-1391
    • Neǐshtadt, A.I.1
  • 25
    • 0000190869 scopus 로고
    • Persistence of stability loss for dynamical bifurcations II
    • A.I. Neǐshtadt, Persistence of stability loss for dynamical bifurcations II, Differential Equations 24 (1988) 171-176.
    • (1988) Differential Equations , vol.24 , pp. 171-176
    • Neǐshtadt, A.I.1
  • 26
    • 0000898198 scopus 로고
    • Asymptotic behavior of solutions of systems of differential equations with a small parameter in the derivatives of highest order
    • L.S. Pontryagin, Asymptotic behavior of solutions of systems of differential equations with a small parameter in the derivatives of highest order, Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957) 605-626.
    • (1957) Izv. Akad. Nauk SSSR. Ser. Mat. , vol.21 , pp. 605-626
    • Pontryagin, L.S.1
  • 27
    • 0001524514 scopus 로고
    • Invariant attracting sets of nonlinear stochastic differential equations
    • H. Langer, V. Nollau (Eds.), Berlin, 1989, Akademie-Verlag, Gaußig
    • B. Schmalfuß, Invariant attracting sets of nonlinear stochastic differential equations, in: H. Langer, V. Nollau (Eds.), Markov Processes and Control Theory, Mathematical Research, Vol. 54, Berlin, 1989, Akademie-Verlag, Gaußig, 1988, pp. 217-228.
    • (1988) Markov Processes and Control Theory, Mathematical Research , vol.54 , pp. 217-228
    • Schmalfuß, B.1
  • 28
    • 0000559545 scopus 로고
    • Systems of differential equations containing small parameters in the derivatives
    • A.N. Tihonov, Systems of differential equations containing small parameters in the derivatives, Mat. Sbornik N.S. 31 (1952) 575-586.
    • (1952) Mat. Sbornik N.S. , vol.31 , pp. 575-586
    • Tihonov, A.N.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.