Harnack-Type Inequalities and Applications for SDE Driven by Fractional Brownian Motion

References

• Aida, S., and Kawabi, H. 2001. Short time asymptotics of a certain infinite dimensional diffusion process. Stochastics Analysis and Related Topics 48:77–124.
   Google Scholar
• Aida, S., and Zhang, T. 2002. On the small time asymptotics of diffusion processes on path groups. Potential Anal. 16:67–78.
   Web of Science ®Google Scholar
• Alòs, E., Mazet, O., and Nualart, D. 2001. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29:766–801.
   Web of Science ®Google Scholar
• Biagini, F., Hu, Y., Øksendal, B., and Zhang, T. 2008. Stochastic Calculus for Fractional Brownian Motion and Applications. Springer-Verlag, London.
   Google Scholar
• Bobkov, S., Gentil, I., and Ledoux, M. 2001. Hypercontractivity of Hamilton- Jacobi equations. J. Math. Pures Appl. 80:669–696.
   Web of Science ®Google Scholar
• Coutin, L., and Qian, Z. 2002. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122:108–140.
   Web of Science ®Google Scholar
• Decreusefond, L., and Üstünel, A.S. 1998. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10:177–214.
   Web of Science ®Google Scholar
• Fan, X. L. 2012. Derivative formula, integration by parts formula and applications for SDEs driven by fractional Brownian motion. arXiv:1206.0961.
   Google Scholar
• Fan, X. L. 2013. Derivative formulae and Harnack inequalities for SDEs with fractional noises. . arXiv:1308.5309.
   Google Scholar
• Fan, X.L. 2013. Harnack inequality and derivative formula for SDE driven by fractional Brownian motion. Science in China-Mathematics 561:515–524.
   Web of Science ®Google Scholar
• Gong, F., and Wang, F.Y. 2001. Heat kernel estimates with application to compactness of manifolds. Q. J. Math. 52:171–180.
   Web of Science ®Google Scholar
• Guillin, A., and Wang, F.Y. 2012. Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality. J. Differential Equations 253:20–40.
   Web of Science ®Google Scholar
• Kawabi, H. 2005. The parabolic Harnack inequality for the time dependent Ginzburg-Landau type SPDE and its application. Potential Anal. 22:61–84.
   Web of Science ®Google Scholar
• Liu, W. Harnack inequality and applications for stochastic evolution equations with monotone drifts. 2009. J. Evol. Equ. 9:747–770.
   Web of Science ®Google Scholar
• Lyons, T. 1998. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14:215–310.
   Web of Science ®Google Scholar
• Nikiforov, A.F., and Uvarov, V.B. 1988. Special Functions of Mathematical Physics. Birkhäuser, Boston.
   Google Scholar
• Nualart, D. 2006. The Malliavin Calculus and Related Topics, 2nd ed. Springer-Verlag, Berlin.
   Google Scholar
• Nualart, D., and Ouknine, Y. 2002. Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102:103–116.
   Web of Science ®Google Scholar
• Nualart, D., and Răşcanu, A. 2002. Differential equations driven by fractional Brownian motion. Collect. Math. 53:55–81.
   Google Scholar
• Ouyang, S.X. 2009. Harnack inequalities and applications for stochastic equations. PhD thesis, Bielefeld University.
   Google Scholar
• Ouyang, S.X., Röckner, M., and Wang, F.Y. 2012. Harnack inequalities and applications for Ornstein-Uhlenbeck semigroups with jump. Potential Anal. 36:301–315.
   Web of Science ®Google Scholar
• Da Prato, G., Röckner, M., and Wang, F.Y. 2009. Singular stochastic equations on Hilberts space: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257:992–1017.
   Web of Science ®Google Scholar
• Röckner, M., and Wang, F.Y. 2003. Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203:237–261.
   Web of Science ®Google Scholar
• Röckner, M., and Wang, F.Y. 2003. Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15:893–921.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Samko, S.G., Kilbas, A.A., and Marichev, O.I. 1993. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publishers, Yvendon.
   Google Scholar
• Wang, F.Y. 1997. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109:417–424.
   Web of Science ®Google Scholar
• Wang, F.Y. 1999. Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants. Ann. Probab. 27:653–663.
   Web of Science ®Google Scholar
• Wang, F.Y. 2007. Harnack inequality and applications for stochastic generalized porous media equations. Ann. Probab. 35:1333–1350.
   Web of Science ®Google Scholar
• Wang, F.Y. 2010. Harnack inequalities on manifolds with boundary and applications. J. Math. Pures Appl. 94:304–321.
   Web of Science ®Google Scholar
• Wang, F.Y. 2011. Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds. Ann. Probab. 39:1449–1467.
   Web of Science ®Google Scholar
• Wang, F.Y. 2013. Harnack Inequalities for Stochastic Partial Differential Equations. Springer, New York.
   Google Scholar
• Wang, F.Y., and Yuan, C. 2011. Harnack inequalities for functional SDEs with multiplicative noise and applications. Stochastic Process. Appl. 121:2692–2710.
   Web of Science ®Google Scholar
• Young, L.C. 1936. An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67:251–282.
   Google Scholar
• Zähle, M. 1998. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111:333–374.
   Web of Science ®Google Scholar
• Zhang, X.C. 2013. Derivative formulas and gradient estimates for SDEs driven by α-stable processes. Stochastic Process. Appl. 123:1213–1228.
   Web of Science ®Google Scholar