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June  2022, 7(2): 119-132. doi: 10.3934/puqr.2022008

Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  December 16, 2021 Accepted  June 07, 2022 Published  June 2022 Early access  June 2022

Fund Project: The authors would like to thank Doctor Xing Huang for corrections and helpful comments. This work was supported in part by NNSFC (Grant No. 12101390).

In this paper, the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup is obtained. All of the above results extend the existed results in linear expectation setting.

Citation: Fen-Fen Yang. Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 119-132. doi: 10.3934/puqr.2022008
References:
[1]

Aida, S., Uniform positivity improving property, Sobolev inequalities, and spectral gaps, J. Funct. Anal., 1998, 158(1): 152−185. doi: 10.1006/jfan.1998.3286.

[2]

Aida, S. and Kawabi, H., Short time asymptotics of certain infinite dimensional diffusion process, In: Decreusefond, L., Øksendal, B. K. and Üstünel, A. S.(eds), Stochastic Analysis and Related Topics, VII, Progress in Probability, Springer, 2001, 48: 77−124.

[3]

Aida, S. and Zhang, T., On the small time asymptotics of diffusion processes on path groups, Potential Anal., 2002, 16: 67−78. doi: 10.1023/A:1024868720071.

[4]

Bao, J., Wang, F.-Y. and Yuan, C., Derivative formula and Harnack inequality for degenerate functionals SDEs, Stoch. Dyn., 2013, 13(1): 943−951.

[5]

Bobkov, S., Gentil, I. and Ledoux, M., Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl., 2001, 80(7): 669−696. doi: 10.1016/S0021-7824(01)01208-9.

[6]

Denis, L., Hu, M. and Peng, S., Function spaces and capacity related to a sublinear expectation: application to G -Brownian motion paths, Potential Anal., 2011, 34(2): 139−161. doi: 10.1007/s11118-010-9185-x.

[7]

Es-Sarhir, A., von Renesse, M.-K. and Scheutzow, M., Harnack inequality for functional SDEs with bounded memory, Electron. Commun. Probab., 2009, 14: 560−565.

[8]

Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by G -Brownian motion, Stochastic Process. Appl., 2009, 119(10): 3356−3382. doi: 10.1016/j.spa.2009.05.010.

[9]

Gong, F. and Wang, F.-Y., Heat kernel estimates with application to compactness of manifolds, Q. J. Math., 2001, 52: 171−180. doi: 10.1093/qjmath/52.2.171.

[10]

Guillin, A. and Wang, F.-Y., Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differ. Equ., 2012, 253(1): 20−40. doi: 10.1016/j.jde.2012.03.014.

[11]

Hu, M. and Ji, S., Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity, SIAM J. Control Optim., 2016, 54(2): 918−945. doi: 10.1137/15M1037639.

[12]

Hu, M., Ji, S., Peng, S. and Song, Y., Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G -Brownian motion, Stochastic Process. Appl., 2014, 124(2): 1170−1195. doi: 10.1016/j.spa.2013.10.009.

[13]

Hu, M. and Peng, S., On representation theorem of G -expectations and paths of G -Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25(3): 539−546. doi: 10.1007/s10255-008-8831-1.

[14]

Huang, X. and Yang, F.-F., Harnack inequality and gradient estimate for G -SDEs with degenerate noise, Sci. China Math., 2022, 65(4): 813−826. doi: 10.1007/s11425-020-1784-0.

[15]

Kawabi, H., The parabolic Harnack inequality for the time dependent Ginzburg–Landau type SPDE and its application, Potential Anal., 2005, 22(1): 61−84. doi: 10.1007/s11118-004-6456-4.

[16]

Osuka, E., Girsanov’s formula for G-Brownian motion, Stochastic Process. Appl., 2013, 123(4): 1301−1318. doi: 10.1016/j.spa.2012.12.009.

[17]

Peng, S., G-Brownian motion and dynamic risk measures under volatility uncertainty, arXiv: 0711.2834, 2007.

[18]

Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Benth, F. E., Di Nunno, G., Lindstrøm, T., Øksendal, B., and Zhang, T. (eds), Stoch. Anal. Appl., Springer, Berlin, Heidelberg, 2007, 2: 541–567.

[19]

Peng, S., Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, arXiv: 1002.4546, 2010.

[20]

Ren, Y., Bi, Q. and Sakthivel, R., Stochastic functional differential equations with infinite delay driven by G -Brownian motion, Math. Methods Appl. Sci., 2013, 36(13): 1746−1759. doi: 10.1002/mma.2720.

[21]

Röckner, M. and Wang, F.-Y., Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal., 2003, 203(1): 237−261. doi: 10.1016/S0022-1236(03)00165-4.

[22]

Röckner, M. and Wang, F.-Y., Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds, Forum Math., 2003, 15(6): 893−921.

[23]

Song, Y., Gradient estimates for nonlinear diffusion semigroups by coupling methods, Sci. China Math., 2021, 64(5): 1093−1108. doi: 10.1007/s11425-018-9541-6.

[24]

Wang, F.-Y., Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2000, 3(2): 263−295. doi: 10.1142/S0219025700000194.

[25]

Wang, F.-Y., Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants, Ann. Probab., 1999, 27(2): 653−663.

[26]

Wang, F. -Y., Harnack Inequalities for Stochastic Partial Differential Equations, Springer Briefs in Mathematics, Springer, New York, 2013.

[27]

Wang, F.-Y., Hypercontractivity and applications for stochastic Hamiltonian systems, J. Funct. Anal., 2017, 272(12): 5360−5383. doi: 10.1016/j.jfa.2017.03.015.

[28]

Wang, F.-Y., Logarithmic Sobolev inequalities: conditions and counterexamples, J. Operator Theory, 2001, 46(1): 183−197.

[29]

Wang, F.-Y. and Yuan, C., Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 2011, 121(11): 2692−2710. doi: 10.1016/j.spa.2011.07.001.

[30]

Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Relat. Fields, 1997, 109(3): 417−424. doi: 10.1007/s004400050137.

[31]

Wang, F.-Y. and Zhang, X. C., Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl., 2013, 99(6): 726−740. doi: 10.1016/j.matpur.2012.10.007.

[32]

Yang, F.-F., Harnack and log Harnack inequalities for G-SDEs with multiplicative noise, arXiv: 1907.02317, 2019.

[33]

Yang, F.-F., Harnack inequality and applications for SDEs driven by G -Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2020, 36(3): 627−635. doi: 10.1007/s10255-020-0957-9.

show all references

References:
[1]

Aida, S., Uniform positivity improving property, Sobolev inequalities, and spectral gaps, J. Funct. Anal., 1998, 158(1): 152−185. doi: 10.1006/jfan.1998.3286.

[2]

Aida, S. and Kawabi, H., Short time asymptotics of certain infinite dimensional diffusion process, In: Decreusefond, L., Øksendal, B. K. and Üstünel, A. S.(eds), Stochastic Analysis and Related Topics, VII, Progress in Probability, Springer, 2001, 48: 77−124.

[3]

Aida, S. and Zhang, T., On the small time asymptotics of diffusion processes on path groups, Potential Anal., 2002, 16: 67−78. doi: 10.1023/A:1024868720071.

[4]

Bao, J., Wang, F.-Y. and Yuan, C., Derivative formula and Harnack inequality for degenerate functionals SDEs, Stoch. Dyn., 2013, 13(1): 943−951.

[5]

Bobkov, S., Gentil, I. and Ledoux, M., Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl., 2001, 80(7): 669−696. doi: 10.1016/S0021-7824(01)01208-9.

[6]

Denis, L., Hu, M. and Peng, S., Function spaces and capacity related to a sublinear expectation: application to G -Brownian motion paths, Potential Anal., 2011, 34(2): 139−161. doi: 10.1007/s11118-010-9185-x.

[7]

Es-Sarhir, A., von Renesse, M.-K. and Scheutzow, M., Harnack inequality for functional SDEs with bounded memory, Electron. Commun. Probab., 2009, 14: 560−565.

[8]

Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by G -Brownian motion, Stochastic Process. Appl., 2009, 119(10): 3356−3382. doi: 10.1016/j.spa.2009.05.010.

[9]

Gong, F. and Wang, F.-Y., Heat kernel estimates with application to compactness of manifolds, Q. J. Math., 2001, 52: 171−180. doi: 10.1093/qjmath/52.2.171.

[10]

Guillin, A. and Wang, F.-Y., Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differ. Equ., 2012, 253(1): 20−40. doi: 10.1016/j.jde.2012.03.014.

[11]

Hu, M. and Ji, S., Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity, SIAM J. Control Optim., 2016, 54(2): 918−945. doi: 10.1137/15M1037639.

[12]

Hu, M., Ji, S., Peng, S. and Song, Y., Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G -Brownian motion, Stochastic Process. Appl., 2014, 124(2): 1170−1195. doi: 10.1016/j.spa.2013.10.009.

[13]

Hu, M. and Peng, S., On representation theorem of G -expectations and paths of G -Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25(3): 539−546. doi: 10.1007/s10255-008-8831-1.

[14]

Huang, X. and Yang, F.-F., Harnack inequality and gradient estimate for G -SDEs with degenerate noise, Sci. China Math., 2022, 65(4): 813−826. doi: 10.1007/s11425-020-1784-0.

[15]

Kawabi, H., The parabolic Harnack inequality for the time dependent Ginzburg–Landau type SPDE and its application, Potential Anal., 2005, 22(1): 61−84. doi: 10.1007/s11118-004-6456-4.

[16]

Osuka, E., Girsanov’s formula for G-Brownian motion, Stochastic Process. Appl., 2013, 123(4): 1301−1318. doi: 10.1016/j.spa.2012.12.009.

[17]

Peng, S., G-Brownian motion and dynamic risk measures under volatility uncertainty, arXiv: 0711.2834, 2007.

[18]

Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Benth, F. E., Di Nunno, G., Lindstrøm, T., Øksendal, B., and Zhang, T. (eds), Stoch. Anal. Appl., Springer, Berlin, Heidelberg, 2007, 2: 541–567.

[19]

Peng, S., Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion, arXiv: 1002.4546, 2010.

[20]

Ren, Y., Bi, Q. and Sakthivel, R., Stochastic functional differential equations with infinite delay driven by G -Brownian motion, Math. Methods Appl. Sci., 2013, 36(13): 1746−1759. doi: 10.1002/mma.2720.

[21]

Röckner, M. and Wang, F.-Y., Harnack and functional inequalities for generalized Mehler semigroups, J. Funct. Anal., 2003, 203(1): 237−261. doi: 10.1016/S0022-1236(03)00165-4.

[22]

Röckner, M. and Wang, F.-Y., Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds, Forum Math., 2003, 15(6): 893−921.

[23]

Song, Y., Gradient estimates for nonlinear diffusion semigroups by coupling methods, Sci. China Math., 2021, 64(5): 1093−1108. doi: 10.1007/s11425-018-9541-6.

[24]

Wang, F.-Y., Functional inequalities, semigroup properties and spectrum estimates, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2000, 3(2): 263−295. doi: 10.1142/S0219025700000194.

[25]

Wang, F.-Y., Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constants, Ann. Probab., 1999, 27(2): 653−663.

[26]

Wang, F. -Y., Harnack Inequalities for Stochastic Partial Differential Equations, Springer Briefs in Mathematics, Springer, New York, 2013.

[27]

Wang, F.-Y., Hypercontractivity and applications for stochastic Hamiltonian systems, J. Funct. Anal., 2017, 272(12): 5360−5383. doi: 10.1016/j.jfa.2017.03.015.

[28]

Wang, F.-Y., Logarithmic Sobolev inequalities: conditions and counterexamples, J. Operator Theory, 2001, 46(1): 183−197.

[29]

Wang, F.-Y. and Yuan, C., Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 2011, 121(11): 2692−2710. doi: 10.1016/j.spa.2011.07.001.

[30]

Wang, F.-Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Relat. Fields, 1997, 109(3): 417−424. doi: 10.1007/s004400050137.

[31]

Wang, F.-Y. and Zhang, X. C., Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl., 2013, 99(6): 726−740. doi: 10.1016/j.matpur.2012.10.007.

[32]

Yang, F.-F., Harnack and log Harnack inequalities for G-SDEs with multiplicative noise, arXiv: 1907.02317, 2019.

[33]

Yang, F.-F., Harnack inequality and applications for SDEs driven by G -Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2020, 36(3): 627−635. doi: 10.1007/s10255-020-0957-9.

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