Electronic Journal of Probability

Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients

Jinghai Shao, Feng-Yu Wang, and Chenggui Yuan

Full-text: Open access

Abstract

By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 100, 18 pp.

Dates
Accepted: 23 November 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062422

Digital Object Identifier
doi:10.1214/EJP.v17-2140

Mathematical Reviews number (MathSciNet)
MR3005718

Zentralblatt MATH identifier
1287.60074

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J60: Diffusion processes [See also 58J65] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Keywords
Harnack inequality log-Harnack inequality stochastic (functional) differential equation existence and uniqueness

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Shao, Jinghai; Wang, Feng-Yu; Yuan, Chenggui. Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients. Electron. J. Probab. 17 (2012), paper no. 100, 18 pp. doi:10.1214/EJP.v17-2140. https://projecteuclid.org/euclid.ejp/1465062422


Export citation

References

  • Aida, Shigeki; Kawabi, Hiroshi. Short time asymptotics of a certain infinite dimensional diffusion process. Stochastic analysis and related topics, VII (Kusadasi, 1998), 77–124, Progr. Probab., 48, Birkhäuser Boston, Boston, MA, 2001.
  • Aida, Shigeki; Zhang, Tusheng. On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16 (2002), no. 1, 67–78.
  • Bao, J., Wang, F.-Y. and Yuan, C.: Derivative formula and Harnack inequality for degenerate functional SDEs. to appear in phStochastics and Dynamics.
  • Bobkov, Sergey G.; Gentil, Ivan; Ledoux, Michel. Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80 (2001), no. 7, 669–696.
  • Es-Sarhir, Abdelhadi; von Renesse, Max-K.; Scheutzow, Michael. Harnack inequality for functional SDEs with bounded memory. Electron. Commun. Probab. 14 (2009), 560–565.
  • Fang, Shizan; Zhang, Tusheng. A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields 132 (2005), no. 3, 356–390.
  • Gong, Fu-Zhou; Wang, Feng-Yu. Heat kernel estimates with application to compactness of manifolds. Q. J. Math. 52 (2001), no. 2, 171–180.
  • Guo, H., Philipowski, R. and Thalmaier, A.: An entropy formula for the heat equation on manifolds with time-dependent metric, application to ancient solutions. Preprint.
  • Hofmanová, Martina; Seidler, Jan. On weak solutions of stochastic differential equations. Stoch. Anal. Appl. 30 (2012), no. 1, 100–121.
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3
  • Lan, Guang Qiang. Pathwise uniqueness and non-explosion of SDEs with non-Lipschitzian coefficients. (Chinese) Acta Math. Sinica (Chin. Ser.) 52 (2009), no. 4, 731–736.
  • Liu, Wei; Wang, Feng-Yu. Harnack inequality and strong Feller property for stochastic fast-diffusion equations. J. Math. Anal. Appl. 342 (2008), no. 1, 651–662.
  • Mao, Xuerong. Stochastic differential equations and applications. Second edition. Horwood Publishing Limited, Chichester, 2008. xviii+422 pp. ISBN: 978-1-904275-34-3
  • Röckner, Michael; Wang, Feng-Yu. Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203 (2003), no. 1, 237–261.
  • Röckner, Michael; Wang, Feng-Yu. Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 1, 27–37.
  • Skorohod, A. V. On stochastic differential equations. (Russian) 1962 Proc. Sixth All-Union Conf. Theory Prob. and Math. Statist. (Vilnius, 1960) (Russian) pp. 159–168 Gosudarstv. Izdat. Političesk. i Navčn. Lit. Litovsk. SSR, Vilnius
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4
  • Taniguchi, Takeshi. Successive approximations to solutions of stochastic differential equations. J. Differential Equations 96 (1992), no. 1, 152–169.
  • Taniguchi, Takeshi. The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps. Stochastics 82 (2010), no. 4, 339–363.
  • Truman, Aubrey; Wang, FengYu; Wu, JiangLun; Yang, Wei. A link of stochastic differential equations to nonlinear parabolic equations. Sci. China Math. 55 (2012), no. 10, 1971–1976.
  • Wang, Feng-Yu. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 (1997), no. 3, 417–424.
  • Wang, Feng-Yu. Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants. Ann. Probab. 27 (1999), no. 2, 653–663.
  • Wang, Feng-Yu. Harnack inequalities on manifolds with boundary and applications. J. Math. Pures Appl. (9) 94 (2010), no. 3, 304–321.
  • 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.
  • Wang, Feng-Yu; Yuan, Chenggui. Harnack inequalities for functional SDEs with multiplicative noise and applications. Stochastic Process. Appl. 121 (2011), no. 11, 2692–2710.
  • Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155–167.