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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
References:
[1]

R. Bafico and P. Baldi, Small random perturbations of Peano phenomena, Stochastics, 6 (1982), 279-292. doi: 10.1080/17442508208833208.  Google Scholar

[2]

V. S. Borkar and K. Suresh Kumar, A new markov selection procedure for degenerate diffusions, J. Theor. Probab., 23 (2010), 729-747. doi: 10.1007/s10959-009-0242-6.  Google Scholar

[3]

R. Buckdahn, M. Quincampoix and Y. Ouknine, On limiting values of stochastic differential equations with small noise intensity tending to zero, 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]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, 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]

F. Flandoli, M. Gubinelli and E. Priola, Well posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53. doi: 10.1007/s00222-009-0224-4.  Google Scholar

[7]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon, Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), 555-580. doi: 10.1016/S0246-0203(01)01075-5.  Google Scholar

[8]

S. Herrmann, Phénomène de Peano et grandes déviations, C. R. Acad. Sci. Paris Sér. I, Math., 332 (2001), 1019-1024. doi: 10.1016/S0764-4442(01)01983-8.  Google Scholar

[9]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.  Google Scholar

[10]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z.  Google Scholar

[11]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, 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]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987.  Google Scholar

[13]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979.  Google Scholar

[14]

D. Trevisan, Zero noise limits using local times, Electron. Commun. Probab., 18 (2013), 7 pp. doi: 10.1214/ECP.v18-2587.  Google Scholar

[15]

Y. A. Veretennikov, On strong solution and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb., 39 (1981), 387-403.  Google Scholar

show all references

References:
[1]

R. Bafico and P. Baldi, Small random perturbations of Peano phenomena, Stochastics, 6 (1982), 279-292. doi: 10.1080/17442508208833208.  Google Scholar

[2]

V. S. Borkar and K. Suresh Kumar, A new markov selection procedure for degenerate diffusions, J. Theor. Probab., 23 (2010), 729-747. doi: 10.1007/s10959-009-0242-6.  Google Scholar

[3]

R. Buckdahn, M. Quincampoix and Y. Ouknine, On limiting values of stochastic differential equations with small noise intensity tending to zero, 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]

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, 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]

F. Flandoli, M. Gubinelli and E. Priola, Well posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53. doi: 10.1007/s00222-009-0224-4.  Google Scholar

[7]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon, Ann. Inst. H. Poincaré Probab. Statist., 37 (2001), 555-580. doi: 10.1016/S0246-0203(01)01075-5.  Google Scholar

[8]

S. Herrmann, Phénomène de Peano et grandes déviations, C. R. Acad. Sci. Paris Sér. I, Math., 332 (2001), 1019-1024. doi: 10.1016/S0764-4442(01)01983-8.  Google Scholar

[9]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.  Google Scholar

[10]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z.  Google Scholar

[11]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, 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]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987.  Google Scholar

[13]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979.  Google Scholar

[14]

D. Trevisan, Zero noise limits using local times, Electron. Commun. Probab., 18 (2013), 7 pp. doi: 10.1214/ECP.v18-2587.  Google Scholar

[15]

Y. A. Veretennikov, On strong solution and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb., 39 (1981), 387-403.  Google Scholar

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