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 and Continuous Dynamical Systems, 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]

Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1601-1618. doi: 10.3934/dcdss.2020091

[2]

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

[3]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems and Imaging, 2021, 15 (4) : 599-618. doi: 10.3934/ipi.2021006

[4]

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

[5]

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

[6]

Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076

[7]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375

[8]

Xiao Ai, Guoxi Ni, Tieyong Zeng. Nonconvex regularization for blurred images with Cauchy noise. Inverse Problems and Imaging, 2022, 16 (3) : 625-646. doi: 10.3934/ipi.2021065

[9]

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

[10]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012

[11]

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

[12]

Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241

[13]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014

[14]

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

[15]

Hanwool Na, Myeongmin Kang, Miyoun Jung, Myungjoo Kang. Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Problems and Imaging, 2019, 13 (1) : 117-147. doi: 10.3934/ipi.2019007

[16]

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

[17]

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 and Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[18]

Ignacio Guerra. A semilinear problem with a gradient term in the nonlinearity. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 137-162. doi: 10.3934/dcds.2021110

[19]

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

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (198)
  • HTML views (188)
  • Cited by (0)

Other articles
by authors

[Back to Top]