March 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 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. FlandoliE. 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]

[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]

[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 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. FlandoliE. 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]

[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.

[1]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[2]

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

[3]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[4]

Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62

[5]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355

[6]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

[7]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421

[8]

Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397

[9]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

[10]

Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019

[11]

Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537

[12]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[13]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271

[14]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305

[15]

Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185

[16]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[17]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[18]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

[19]

Bálint Farkas, Luca Lorenzi. On a class of hypoelliptic operators with unbounded coefficients in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1159-1201. doi: 10.3934/cpaa.2009.8.1159

[20]

Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1197-1218. doi: 10.3934/dcds.2011.31.1197

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (24)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]