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

Weak regularization by stochastic drift : Result and counter example

Paul-Eric Chaudru De Raynal

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.

Keywords: Regularization by noise, martingale problem, Kolmogorov hypoelliptic PDEs, parametrix, gradient estimate.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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] 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electronic Journal of Probability, 22 (2017), 42pp.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.   Google Scholar
[14]

M. Hairer, Introduction to regularity structures, Brazilian Journal of Probability and Statistics, 29 (2015), 175-210.  doi: 10.1214/14-BJPS241.  Google Scholar

[15]

L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[16]

A. Kolmogorov, Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)., Ann. of Math., Ⅱ. Ser., 35 (1934), 116-117.  doi: 10.2307/1968123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Menozzi, Martingale problems for some degenerate Kolmogorov equations, Stochastic Processes and their Applications, (2017).  doi: 10.1016/j.spa.2017.06.001.  Google Scholar

[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.  Google Scholar

[22]

A. J. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Matematicheskiĭ Sbornik. Novaya Seriya, 111 (1980), 434-452,480.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electronic Journal of Probability, 22 (2017), 42pp.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.   Google Scholar
[14]

M. Hairer, Introduction to regularity structures, Brazilian Journal of Probability and Statistics, 29 (2015), 175-210.  doi: 10.1214/14-BJPS241.  Google Scholar

[15]

L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[16]

A. Kolmogorov, Zufällige Bewegungen. (Zur Theorie der Brownschen Bewegung.)., Ann. of Math., Ⅱ. Ser., 35 (1934), 116-117.  doi: 10.2307/1968123.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Menozzi, Martingale problems for some degenerate Kolmogorov equations, Stochastic Processes and their Applications, (2017).  doi: 10.1016/j.spa.2017.06.001.  Google Scholar

[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.  Google Scholar

[22]

A. J. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Matematicheskiĭ Sbornik. Novaya Seriya, 111 (1980), 434-452,480.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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