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A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation
Weak regularization by stochastic drift : Result and counter example
Univ. Savoie Mont Blanc, CNRS, LAMA, F-73000 Chambéry, France |
In this paper, weak uniqueness of hypoelliptic stochastic differential equation with Hölder drift is proved when the Hölder exponent is strictly greater than 1/3. This result then "extends" to a weak framework the previous works [
The approach is based on martingale problem formulation of Stroock and Varadhan and so on smoothing properties of the associated PDE which is, in the current setting, degenerate.
References:
[1] |
L. Beck, F. Flandoli, M. Gubinelli and M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, arXiv:1401.1530 [math] Google Scholar |
[2] |
G. Cannizzaro and K. Chouk, Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential, To appear in Annals of Probability, arXiv:1501.04751 [math] Google Scholar |
[3] |
R. Catellier and M. Gubinelli,
Averaging along irregular curves and regularisation of ODEs, Stochastic Processes and their Applications, 126 (2016), 2323-2366.
doi: 10.1016/j.spa.2016.02.002. |
[4] |
P. E. Chaudru de Raynal,
Strong existence and uniqueness for degenerate SDE with Hölder drift, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 53 (2017), 259-286.
doi: 10.1214/15-AIHP716. |
[5] |
F. Delarue and R. Diel,
Rough paths and 1d SDE with a time dependent distributional drift: Application to polymers, Probability Theory and Related Fields, 165 (2016), 1-63.
doi: 10.1007/s00440-015-0626-8. |
[6] |
F. Delarue and F. Flandoli,
The transition point in the zero noise limit for a 1d Peano example, Discrete and Continuous Dynamical Systems, 34 (2014), 4071-4083.
doi: 10.3934/dcds.2014.34.4071. |
[7] |
F. Delarue and S. Menozzi,
Density estimates for a random noise propagating through a chain of differential equations, Journal of Functional Analysis, 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[8] |
M. Di Francesco and S. Polidoro,
Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Advances in Differential Equations, 11 (2006), 1261-1320.
|
[9] |
R. J. DiPerna and P.-L. Lions,
Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[10] |
E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations,
Electronic Journal of Probability, 22 (2017), 42pp. |
[11] |
F. Flandoli, E. Issoglio and F. Russo,
Multidimensional stochastic differential equations with distributional drift, Transactions of the American Mathematical Society, 369 (2017), 1665-1688.
|
[12] |
F. Flandoli,
Random Perturbation of PDEs and Fluid Dynamic Models, vol. 2015 of Lecture Notes in Mathematics, Springer, Heidelberg, 2011, Lectures from the 40th Probability Summer School held in Saint-Flour, 2010. |
[13] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.
![]() |
[14] |
M. Hairer,
Introduction to regularity structures, Brazilian Journal of Probability and Statistics, 29 (2015), 175-210.
doi: 10.1214/14-BJPS241. |
[15] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[16] |
A. Kolmogorov,
Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)., Ann. of Math., Ⅱ. Ser., 35 (1934), 116-117.
doi: 10.2307/1968123. |
[17] |
N. V. Krylov and M. Röckner,
Strong solutions of stochastic equations with singular time dependent drift, Probability Theory and Related Fields, 131 (2005), 154-196.
doi: 10.1007/s00440-004-0361-z. |
[18] |
H. P. McKean Jr. and I. M. Singer,
Curvature and the eigenvalues of the Laplacian, Journal of Differential Geometry, 1 (1967), 43-69.
doi: 10.4310/jdg/1214427880. |
[19] |
S. Menozzi,
Parametrix techniques and martingale problems for some degenerate Kolmogorov equations, Electronic Communications in Probability, 16 (2011), 234-250.
doi: 10.1214/ECP.v16-1619. |
[20] |
S. Menozzi,
Martingale problems for some degenerate Kolmogorov equations, Stochastic Processes and their Applications, (2017).
doi: 10.1016/j.spa.2017.06.001. |
[21] |
D. W. Stroock and S. R. S. Varadhan,
Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1979. |
[22] |
A. J. Veretennikov,
Strong solutions and explicit formulas for solutions of stochastic integral equations, Matematicheskiĭ Sbornik. Novaya Seriya, 111 (1980), 434-452,480.
|
[23] |
F. Wang and X. Zhang,
Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient, SIAM Journal on Mathematical Analysis, 48 (2016), 2189-2226.
doi: 10.1137/15M1023671. |
[24] |
X. Zhang,
Strong solutions of SDES with singular drift and Sobolev diffusion coefficients, Stochastic Processes and their Applications, 115 (2005), 1805-1818.
doi: 10.1016/j.spa.2005.06.003. |
[25] |
X. Zhang, Stochastic Hamiltonian flows with singular coefficients, arXiv:1606.04360 [math] Google Scholar |
[26] |
A. K. Zvonkin,
A transformation of the phase space of a diffusion process that will remove the drift, Mat. Sb. (N.S.), 93 (1974), 129-149,152.
|
show all references
References:
[1] |
L. Beck, F. Flandoli, M. Gubinelli and M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, arXiv:1401.1530 [math] Google Scholar |
[2] |
G. Cannizzaro and K. Chouk, Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential, To appear in Annals of Probability, arXiv:1501.04751 [math] Google Scholar |
[3] |
R. Catellier and M. Gubinelli,
Averaging along irregular curves and regularisation of ODEs, Stochastic Processes and their Applications, 126 (2016), 2323-2366.
doi: 10.1016/j.spa.2016.02.002. |
[4] |
P. E. Chaudru de Raynal,
Strong existence and uniqueness for degenerate SDE with Hölder drift, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 53 (2017), 259-286.
doi: 10.1214/15-AIHP716. |
[5] |
F. Delarue and R. Diel,
Rough paths and 1d SDE with a time dependent distributional drift: Application to polymers, Probability Theory and Related Fields, 165 (2016), 1-63.
doi: 10.1007/s00440-015-0626-8. |
[6] |
F. Delarue and F. Flandoli,
The transition point in the zero noise limit for a 1d Peano example, Discrete and Continuous Dynamical Systems, 34 (2014), 4071-4083.
doi: 10.3934/dcds.2014.34.4071. |
[7] |
F. Delarue and S. Menozzi,
Density estimates for a random noise propagating through a chain of differential equations, Journal of Functional Analysis, 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[8] |
M. Di Francesco and S. Polidoro,
Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Advances in Differential Equations, 11 (2006), 1261-1320.
|
[9] |
R. J. DiPerna and P.-L. Lions,
Ordinary differential equations, transport theory and Sobolev spaces, Inventiones Mathematicae, 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[10] |
E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations,
Electronic Journal of Probability, 22 (2017), 42pp. |
[11] |
F. Flandoli, E. Issoglio and F. Russo,
Multidimensional stochastic differential equations with distributional drift, Transactions of the American Mathematical Society, 369 (2017), 1665-1688.
|
[12] |
F. Flandoli,
Random Perturbation of PDEs and Fluid Dynamic Models, vol. 2015 of Lecture Notes in Mathematics, Springer, Heidelberg, 2011, Lectures from the 40th Probability Summer School held in Saint-Flour, 2010. |
[13] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.
![]() |
[14] |
M. Hairer,
Introduction to regularity structures, Brazilian Journal of Probability and Statistics, 29 (2015), 175-210.
doi: 10.1214/14-BJPS241. |
[15] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[16] |
A. Kolmogorov,
Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)., Ann. of Math., Ⅱ. Ser., 35 (1934), 116-117.
doi: 10.2307/1968123. |
[17] |
N. V. Krylov and M. Röckner,
Strong solutions of stochastic equations with singular time dependent drift, Probability Theory and Related Fields, 131 (2005), 154-196.
doi: 10.1007/s00440-004-0361-z. |
[18] |
H. P. McKean Jr. and I. M. Singer,
Curvature and the eigenvalues of the Laplacian, Journal of Differential Geometry, 1 (1967), 43-69.
doi: 10.4310/jdg/1214427880. |
[19] |
S. Menozzi,
Parametrix techniques and martingale problems for some degenerate Kolmogorov equations, Electronic Communications in Probability, 16 (2011), 234-250.
doi: 10.1214/ECP.v16-1619. |
[20] |
S. Menozzi,
Martingale problems for some degenerate Kolmogorov equations, Stochastic Processes and their Applications, (2017).
doi: 10.1016/j.spa.2017.06.001. |
[21] |
D. W. Stroock and S. R. S. Varadhan,
Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1979. |
[22] |
A. J. Veretennikov,
Strong solutions and explicit formulas for solutions of stochastic integral equations, Matematicheskiĭ Sbornik. Novaya Seriya, 111 (1980), 434-452,480.
|
[23] |
F. Wang and X. Zhang,
Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient, SIAM Journal on Mathematical Analysis, 48 (2016), 2189-2226.
doi: 10.1137/15M1023671. |
[24] |
X. Zhang,
Strong solutions of SDES with singular drift and Sobolev diffusion coefficients, Stochastic Processes and their Applications, 115 (2005), 1805-1818.
doi: 10.1016/j.spa.2005.06.003. |
[25] |
X. Zhang, Stochastic Hamiltonian flows with singular coefficients, arXiv:1606.04360 [math] Google Scholar |
[26] |
A. K. Zvonkin,
A transformation of the phase space of a diffusion process that will remove the drift, Mat. Sb. (N.S.), 93 (1974), 129-149,152.
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