Shift Harnack inequality and integration by parts formula for functional SDEs driven by fractional Brownian motion

Li, Zhi

doi:10.1090/proc/12915

Shift Harnack inequality and integration by parts formula for functional SDEs driven by fractional Brownian motion
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by Zhi Li

Proc. Amer. Math. Soc. 144 (2016), 2651-2659

DOI: https://doi.org/10.1090/proc/12915

Published electronically: December 22, 2015

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Abstract:

The shift Harnack inequality and the integration by parts formula for functional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $\frac 12 <H<1$ are established by using a transformation formula for fractional Brownian motion and a new coupling argument.

References

Elisa Alòs, Olivier Mazet, and David Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), no. 2, 766–801. MR 1849177, DOI 10.1214/aop/1008956692
Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 755001
B. Boufoussi and S. Hajji, Functional differential equations driven by a fractional Brownian motion, Comput. Math. Appl. 62 (2011), no. 2, 746–754. MR 2817911, DOI 10.1016/j.camwa.2011.05.055
Bruce K. Driver, Integration by parts for heat kernel measures revisited, J. Math. Pures Appl. (9) 76 (1997), no. 8, 703–737. MR 1470881, DOI 10.1016/S0021-7824(97)89966-7
L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), no. 2, 177–214. MR 1677455, DOI 10.1023/A:1008634027843
K. D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1994), no. 1, 252–286. MR 1297021, DOI 10.1006/jfan.1994.1124
XiLiang Fan, Harnack inequality and derivative formula for SDE driven by fractional Brownian motion, Sci. China Math. 56 (2013), no. 3, 515–524. MR 3017935, DOI 10.1007/s11425-013-4569-1
Xiliang Fan, Integration by parts formula and applications for SDEs driven by fractional Brownian motions, Stoch. Anal. Appl. 33 (2015), no. 2, 199–212. MR 3305466, DOI 10.1080/07362994.2014.975819
Céline Jost, Transformation formulas for fractional Brownian motion, Stochastic Process. Appl. 116 (2006), no. 10, 1341–1357. MR 2260738, DOI 10.1016/j.spa.2006.02.006
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
David Nualart and Youssef Ouknine, Regularization of differential equations by fractional noise, Stochastic Process. Appl. 102 (2002), no. 1, 103–116. MR 1934157, DOI 10.1016/S0304-4149(02)00155-2
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Feng-Yu Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. (9) 94 (2010), no. 3, 304–321 (English, with English and French summaries). MR 2679029, DOI 10.1016/j.matpur.2010.03.001
Feng-Yu Wang, Integration by parts formula and shift Harnack inequality for stochastic equations, Ann. Probab. 42 (2014), no. 3, 994–1019. MR 3189064, DOI 10.1214/13-AOP875
Feng-Yu Wang and Xi-Cheng Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl. (9) 99 (2013), no. 6, 726–740 (English, with English and French summaries). MR 3055216, DOI 10.1016/j.matpur.2012.10.007
Xicheng Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl. 120 (2010), no. 10, 1929–1949. MR 2673982, DOI 10.1016/j.spa.2010.05.015
S. Q. Zhang, Shift Harnack inequality and integration by part formula for semilinear SPDE, arXiv:1208.2425v3, 2012.