American Institute of Mathematical Sciences

Advanced
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

[1]

Stefano Galatolo, Hugo Marsan. Quadratic response and speed of convergence of invariant measures in the zero-noise limit. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5303-5327. doi: 10.3934/dcds.2021078
[2]

Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305
[3]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013
[4]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006
[5]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[6]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
[7]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[8]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[9]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[10]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[11]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
[12]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
[13]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375
[14]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
[15]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[16]

Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
[17]

Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021133
[18]

Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021207
[19]

Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021204
[20]

Yanqiang Chang, Huabin Chen. Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021301

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy