Annals of Probability

Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds

Feng-Yu Wang

Full-text: Open access


By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.

Article information

Ann. Probab., Volume 39, Number 4 (2011), 1449-1467.

First available in Project Euclid: 5 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Harnack inequality stochastic differential equation Neummann semigroup manifold


Wang, Feng-Yu. Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds. Ann. Probab. 39 (2011), no. 4, 1449--1467. doi:10.1214/10-AOP600.

Export citation


  • [1] Aida, S. (1998). Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 152–185.
  • [2] Aida, S. and Kawabi, H. (2001). Short time asymptotics of a certain infinite dimensional diffusion process. In Stochastic Analysis and Related Topics, VII (Kusadasi, 1998). Progress in Probability 48 77–124. Birkhäuser, Boston, MA.
  • [3] Aida, S. and Zhang, T. (2002). On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16 67–78.
  • [4] Arnaudon, M., Thalmaier, A. and Wang, F.-Y. (2006). Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130 223–233.
  • [5] Bakry, D. and Émery, M. (1984). Hypercontractivité de semi-groupes de diffusion. C. R. Acad. Sci. Paris Sér. I Math. 299 775–778.
  • [6] Bogachev, V. I., Krylov, N. V. and Röckner, M. (2001). On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Partial Differential Equations 26 2037–2080.
  • [7] Chen, X. and Wang, F.-Y. (2007). Optimal integrability condition for the log-Sobolev inequality. Q. J. Math. 58 17–22.
  • [8] Da Prato, G., Röckner, M. and Wang, F.-Y. (2009). Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257 992–1017.
  • [9] Émery, M. (1989). Stochastic Calculus in Manifolds. Springer, Berlin.
  • [10] Es-Sarhir, A., von Renesse, M.-K. and Scheutzew, M. (2009). Harnack inequality for functional SDEs with bounded memory. Electron. Comm. Probab. 14 560–565.
  • [11] Fang, S. and Zhang, T. (2005). A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields 132 356–390.
  • [12] Hino, M. (2000). Exponential decay of positivity preserving semigroups on Lp. Osaka J. Math. 37 603–624.
  • [13] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [14] Liu, W. (2009). Fine properties of stochastic evolution equations and their applications. Doctor-thesis, Bielefeld Univ.
  • [15] Liu, W. and Wang, F.-Y. (2008). Harnack inequality and strong Feller property for stochastic fast-diffusion equations. J. Math. Anal. Appl. 342 651–662.
  • [16] Ouyang, S. X. (2009). Harnack inequalities and applications for stochastic equations. Ph.D. thesis, Bielefeld Univ.
  • [17] Röckner, M. and Wang, F.-Y. (2003). Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203 237–261.
  • [18] Röckner, M. and Wang, F.-Y. (2003). Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15 893–921.
  • [19] Röckner, M. and Wang, F.-Y. (2010). Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 27–37.
  • [20] Thalmaier, A. and Wang, F.-Y. (1998). Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155 109–124.
  • [21] Wang, F. Y. (2009). Transportation-cost inequalities on path space over manifolds with boundary. Preprint. Available at arXiv:0908.2891.
  • [22] Wang, F.-Y. (1997). Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 417–424.
  • [23] Wang, F.-Y. (2004). Equivalence of dimension-free Harnack inequality and curvature condition. Integral Equations Operator Theory 48 547–552.
  • [24] Wang, F.-Y. (2007). Estimates of the first Neumann eigenvalue and the log-Sobolev constant on non-convex manifolds. Math. Nachr. 280 1431–1439.
  • [25] Wang, F.-Y. (2007). Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35 1333–1350.
  • [26] Wang, F.-Y. (2010). Harnack inequalities on manifolds with boundary and applications. J. Math. Pures Appl. (9) 94 304–321.
  • [27] Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155–167.
  • [28] Zhang, T. S. (2010). White noise driven SPDEs with reflection: Strong Feller properties and Harnack inequalities. Potential Analysis 33 137–151.