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October  2014, 34(10): 4071-4083. doi: 10.3934/dcds.2014.34.4071

The transition point in the zero noise limit for a 1D Peano example

François Delarue 1, and  Franco Flandoli 2,

1. 

Laboratoire J.A. Dieudonné, UMR CNRS-UNS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France
2. 

Dipartimento Matematica, Via Buonarroti 1c, C.A.P. 56127, Pisa, Italy

Received  February 2013 Revised  April 2013 Published  April 2014

The zero-noise result for Peano phenomena of Bafico and Baldi (1982) is revisited. The original proof was based on explicit solutions to the elliptic equations for probabilities of exit times. The new proof given here is purely dynamical, based on a direct analysis of the SDE and the relative importance of noise and drift terms. The transition point between noisy behavior and escaping behavior due to the drift is identified.
Keywords: Stochastic differential equations, zero-noise limit, Peano phenomena, singular drifts, non-uniqueness..
Mathematics Subject Classification: Primary: 60H10, 34A12; Secondary: 60F99, 60J6.
Citation: François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071
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show all references

References:
[1]

Stochastics, 6 (1982), 279-292. doi: 10.1080/17442508208833208.  Google Scholar

[2]

J. Theor. Probab., 23 (2010), 729-747. doi: 10.1007/s10959-009-0242-6.  Google Scholar

[3]

Bull. Sci. Math., 133 (2009), 229-237. doi: 10.1016/j.bulsci.2008.12.005.  Google Scholar

[4]

F. Delarue, F. Flandoli and D. Vincenzi, Noise prevents collapse of Vlasov-Poisson point charges,, to appear in Comm. Pure Appl. Math., ().  doi: 10.1002/cpa.21476.  Google Scholar

[5]

Lectures from the 40th Probability Summer School held in Saint-Flour, 2010, Lecture Notes in Mathematics, 2015, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18231-0.  Google Scholar

[6]

Invent. Math., 180 (2010), 1-53. doi: 10.1007/s00222-009-0224-4.  Google Scholar

[7]

Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), 555-580. doi: 10.1016/S0246-0203(01)01075-5.  Google Scholar

[8]

C. R. Acad. Sci. Paris Sér. I, Math., 332 (2001), 1019-1024. doi: 10.1016/S0764-4442(01)01983-8.  Google Scholar

[9]

Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.  Google Scholar

[10]

Probab. Theory Related Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z.  Google Scholar

[11]

Ecole d'été de probabilités de Saint-Flour, XII-1982, Lecture Notes in Math., 1097, Springer, Berlin, 1984, 143-303. doi: 10.1007/BFb0099433.  Google Scholar

[12]

Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987.  Google Scholar

[13]

Springer-Verlag, Berlin 1979.  Google Scholar

[14]

Electron. Commun. Probab., 18 (2013), 7 pp. doi: 10.1214/ECP.v18-2587.  Google Scholar

[15]

Math. USSR Sb., 39 (1981), 387-403.  Google Scholar

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