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The behavior of a beam fixed on small sets of one of its extremities
The transition point in the zero noise limit for a 1D Peano example
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 |
References:
[1] |
R. Bafico and P. Baldi, Small random perturbations of Peano phenomena, Stochastics, 6 (1982), 279-292.
doi: 10.1080/17442508208833208. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979. |
[14] |
D. Trevisan, Zero noise limits using local times, Electron. Commun. Probab., 18 (2013), 7 pp.
doi: 10.1214/ECP.v18-2587. |
[15] |
Y. A. Veretennikov, On strong solution and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb., 39 (1981), 387-403. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979. |
[14] |
D. Trevisan, Zero noise limits using local times, Electron. Commun. Probab., 18 (2013), 7 pp.
doi: 10.1214/ECP.v18-2587. |
[15] |
Y. A. Veretennikov, On strong solution and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb., 39 (1981), 387-403. |
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