# American Institue of Mathematical Sciences

2018, 38(3): 1269-1291. doi: 10.3934/dcds.2018052

## Weak regularization by stochastic drift : Result and counter example

 Univ. Savoie Mont Blanc, CNRS, LAMA, F-73000 Chambéry, France

Received  March 2017 Revised  September 2017 Published  December 2017

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 [4,23,10], where strong uniqueness was proved when the regularity index of the drift is strictly greater than 2/3. Part of the result is also shown to be almost sharp thanks to a counter example when the Hölder exponent of the degenerate component is just below 1/3.

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.

Citation: Paul-Eric Chaudru De Raynal. Weak regularization by stochastic drift : Result and counter example. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1269-1291. doi: 10.3934/dcds.2018052
##### 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] [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] [3] R. Catellier, 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, 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, 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, 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, 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, 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, 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, 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., 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, 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] [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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##### 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] [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] [3] R. Catellier, 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, 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, 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, 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, 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, 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, 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, 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., 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, 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] [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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