Startseite Harnack-type inequality for linear fractional stochastic equations
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Harnack-type inequality for linear fractional stochastic equations

  • Brahim Boufoussi und Soufiane Mouchtabih EMAIL logo
Veröffentlicht/Copyright: 7. November 2020
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter H<12. We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.

MSC 2010: 60H10; 60G22

Communicated by Anatoly F. Turbin


References

[1] S. G. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton–Jacobi equations, J. Math. Pures Appl. 80 (2001), 669–696. 10.1016/S0021-7824(01)01208-9Suche in Google Scholar

[2] O. El Barrimi and Y. Ouknine, Stochastic differential equations driven by an additive fractional Brownian sheet, Bull. Korean Math. Soc. 56 (2019), no. 2, 479–489. Suche in Google Scholar

[3] M. Erraoui, Y. Ouknine and D. Nualart, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stoch. Dyn. 3 (2003), no. 2, 121–139. 10.1142/S0219493703000681Suche in Google Scholar

[4] X. L. Fan, Harnack inequality and derivative formula for SDE driven by fractional Brownian motion, Sci. China Math. 561 (2013), 515–524. 10.1007/s11425-013-4569-1Suche in Google Scholar

[5] X. L. Fan, Harnack-type inequalities and applications for sde driven by fractional Brownian motion, Stoch. Anal. Appl. 32 (2014), no. 4, 602–618. 10.1080/07362994.2014.907745Suche in Google Scholar

[6] F. Z. Gong and F. Y. Wang, Heat kernel estimates with applications to compactness of manifolds, Quart. J. Math. 25 (2001), 171–180. 10.1093/qjmath/52.2.171Suche in Google Scholar

[7] X. Huang, Harnack and shift Harnack inequalities for SDEs with integrable drifts, Stoch. Dyn. 19 (2019), 10.1142/S0219493719500345. 10.1142/S0219493719500345Suche in Google Scholar

[8] W. Liu and F. Y. Wang, Harnack inequality and strong Feller property for stochastic fast-diffusion equations, J. Math. Anal. Appl. 342 (2008), 651–662. 10.1016/j.jmaa.2007.12.047Suche in Google Scholar

[9] D. Nualart and Y. Ouknine, Regularization of differential equations by fractional noise, Stochastic Process. Appl. 102 (2002), 103–116. 10.1016/S0304-4149(02)00155-2Suche in Google Scholar

[10] M. Rockner and F. Y. Wang, Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal. 203 (2003), 237–261. 10.1016/S0022-1236(03)00165-4Suche in Google Scholar

[11] M. Rockner and F. Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds, Forum Math. 15 (2003), 893–921. Suche in Google Scholar

[12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science, Yvendon, 1993. Suche in Google Scholar

[13] J. Shao, F. Y. Wang and C. Yuan, Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients, Electron. J. Probab. 17 (2011), no. 100, 1–18. 10.1214/EJP.v17-2140Suche in Google Scholar

[14] F. Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997), 417–424. 10.1007/s004400050137Suche in Google Scholar

[15] F. Y. Wang, Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab. 27 (1999), 653–663. 10.1214/aop/1022677381Suche in Google Scholar

[16] F. Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab. 39 (2011), no. 4, 1449–1467. 10.1214/10-AOP600Suche in Google Scholar

[17] F. Y. Wang, J. L. Wu and L. H. Xu, Log-Harnack inequality for stochastic Burgers equations and applications, J. Math.Anal. Appl. 384 (2011), 151–159. 10.1016/j.jmaa.2011.02.032Suche in Google Scholar

[18] X. Zhang, Exponential ergodicity of non-Lipschitz stochastic differential equations, Proc. Amer. Math. Soc. 137 (2009), 317–327. 10.1090/S0002-9939-08-09509-9Suche in Google Scholar

Received: 2020-02-20
Accepted: 2020-09-29
Published Online: 2020-11-07
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 3.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/rose-2020-2046/html?lang=de
Button zum nach oben scrollen