Electronic Journal of Probability

Harnack inequalities for SDEs driven by time-changed fractional Brownian motions

Chang-Song Deng and René L. Schilling

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We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in (0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition; otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 71, 23 pp.

Received: 4 September 2016
Accepted: 11 July 2017
First available in Project Euclid: 13 September 2017

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Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60H10: Stochastic ordinary differential equations [See also 34F05] 60G15: Gaussian processes

Harnack inequality fractional Brownian motion random time-change stochastic differential equation

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Deng, Chang-Song; Schilling, René L. Harnack inequalities for SDEs driven by time-changed fractional Brownian motions. Electron. J. Probab. 22 (2017), paper no. 71, 23 pp. doi:10.1214/17-EJP82. https://projecteuclid.org/euclid.ejp/1505268102

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  • [1] Arnaudon, M., Thalmaier, A., and Wang, F.-Y.: Equivalent Harnack and gradient inequalities for pointwise curvature lower bound. Bull. Sci. Math. 138, (2014), 643–655.
  • [2] Biagini, F., Hu, Y., Øksendal, B. and Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London 2008.
  • [3] Bihari, I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Hung. 7, (1956), 81–94.
  • [4] Da Prato, G., Röckner, M., and Wang, F.-Y.: Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257, (2009), 992–1017.
  • [5] Decreusefond, L. and Üstünel, A. S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, (1999), 177–214.
  • [6] Deng, C.-S.: Harnack inequalities for SDEs driven by subordinate Brownian motions. J. Math. Anal. Appl. 417, (2014), 970–978.
  • [7] Deng, C.-S. and Schilling, R.L.: On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes. Stochastic Process. Appl. 125, (2015), 3851–3878.
  • [8] Fan, X.-L.: Harnack inequality and derivative formula for SDE driven by fractional Brownian motion. Sci. China Math. 56, (2013), 515–524.
  • [9] Fan, X.-L.: Harnack-type inequalities and applications for SDE driven by fractional Brownian motion. Stoch. Anal. Appl. 32, (2014), 602–618.
  • [10] Fan, X.-L. and Ren, Y.: Bismut formulas and applications for stochastic (functional) differential equations driven by fractional Brownian motions. Stoch. Dyn. 17, (2017), 1750028, 19 pages.
  • [11] Gajda, J. and Magdziarz, M.: Large deviations for subordinated Brownian motion and applications. Statist. Probab. Lett. 88, (2014), 149–156.
  • [12] Linde, W. and Shi, Z.: Evaluating the small deviation probabilities for subordinated Lévy processes. Stochastic Process. Appl. 113, (2004), 273–287.
  • [13] Meerschaert, M.M., Nane, E., and Xiao, Y.: Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, (2008), 432–445.
  • [14] Nourdin, I.: Selected Aspects of Fractional Brownian Motion. Springer, Milan 2012.
  • [15] Nualart, D. and Ouknine, Y.: Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102, (2002), 103–116.
  • [16] Röckner, M. and Wang, F.-Y.: Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Inf. Dim. Anal. Quantum Probab. Rel. Top. 13, (2010), 27–37.
  • [17] Samko, S.G., Kilbas, A.A., and Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publishers, 1993.
  • [18] Schilling, R.L., Song, R., and Vondraček, Z.: Bernstein Functions. Theory and Applications (2nd edn). De Gruyter, Studies in Mathematics 37, Berlin 2012.
  • [19] Wang, F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109, (1997), 417–424.
  • [20] Wang, F.-Y.: Harnack Inequalities for Stochastic Partial Differential Equations. Springer, New York 2013.
  • [21] Wang, F.-Y. and Wang, J.: Harnack inequalities for stochastic equations driven by Lévy noise. J. Math. Anal. Appl. 410, (2014), 513–523.
  • [22] Wang, L. and Zhang, X.: Harnack inequalities for SDEs driven by cylindrical $\alpha $-stable processes. Potential Anal. 42, (2015), 657–669.
  • [23] Zhang, X.: Derivative formula and gradient estimates for SDEs driven by $\alpha $-stable processes. Stochastic Process. Appl. 123, (2013), 1213–1228.