Harnack inequalities for functional SDEs driven by subordinate Volterra-Gaussian processes

References

• Alòs, E., Mazet, O., Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29: 766–801.
   Google Scholar
• Čoupek, P., Maslowski, B. (2017). Stochastic evolution equations with Volterra noise. Stoch. Process. Appl. 127(3): 877–900. doi:10.1016/j.spa.2016.07.003.
   Web of Science ®Google Scholar
• Čoupek, P., Maslowski, B., Ondreját, M. (2018). Lp-valued stochastic convolution integral driven by Volterra noise. Stoch. Dyn. 18(06): 1850048. doi:10.1142/S021949371850048X.
   Google Scholar
• Čoupek, P. (2018). Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise. Stoch. Anal. Appl. 36(3): 393–412. doi:10.1080/07362994.2017.1409124.
   Web of Science ®Google Scholar
• Duncan, T. E., Maslowski, B., Pasik-Duncan, B. (2019). Linear stochastic differential equations driven by Gauss-Volterra processes and related linear-quadratic control problems. Appl. Math. Optim. 80(2): 369–389. doi:10.1007/s00245-017-9468-3.
   Web of Science ®Google Scholar
• Kumar, V., Mohan, M. T., Kumar Giri, A. (2022). On a generalized stochastic Burgers’ equation perturbed by Volterra noise. J. Math. Anal. Appl. 506(1): 125638. doi:10.1016/j.jmaa.2021.125638.
   Web of Science ®Google Scholar
• Cass, T., Lim, N. (2021). Skorohod and rough integration for stochastic differential equations driven by Volterra processes. Ann. Inst. H. Poincar Probab. Statist. 57(1): 132–168.
   Web of Science ®Google Scholar
• Wang, F.-Y. (1997). Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields. 109(3): 417–424. doi:10.1007/s004400050137.
   Web of Science ®Google Scholar
• Röckner, M., Wang, F.-Y. (2010). Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 13(01): 27–37. doi:10.1142/S0219025710003936.
   Google Scholar
• Röckner, M., Wang, F.-Y. (2003). Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203(1): 237–261. doi:10.1016/S0022-1236(03)00165-4.
   Web of Science ®Google Scholar
• Röckner, M., Wang, F. Y. (2003). Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15: 893–921.
   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
• Gong, F.-Z. (2001). Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52(2): 171–180. doi:10.1093/qjmath/52.2.171.
   Google Scholar
• Shao, J. (2013). Harnack inequalities and heat kernel estimates for SDEs with singular drifts. Bull. Sci. Math. 137(5): 589–601. doi:10.1016/j.bulsci.2012.12.003.
   Web of Science ®Google Scholar
• Wang, W. (2020). Harnack inequality, heat kernel bounds and eigenvalue estimates under integral Ricci curvature bounds. J. Differ. Equ. 269(2): 1243–1277. doi:10.1016/j.jde.2020.01.003.
   Web of Science ®Google Scholar
• Wang, W., Zhang, P. (2017). Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds. Math. Nachr. 290(11-12): 1905–1917. doi:10.1002/mana.201500287.
   Web of Science ®Google Scholar
• Bao, J., Wang, F.-Y., Yuan, C. (2019). Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory. Stoch. Process. Appl. 129(11): 4576–4596. doi:10.1016/j.spa.2018.12.010.
   Web of Science ®Google Scholar
• Niu, M., Xie, B. (2019). Wang’s Harnack inequalities for space–time white noises driven SPDEs with two reflecting walls and their applications. J. Math. Anal. Appl. 469(2): 568–593. doi:10.1016/j.jmaa.2018.09.029.
   Web of Science ®Google Scholar
• Wang, F.-Y., Yuan, C. (2011). Harnack inequalities for functional SDEs with multiplicative noise and applications. Stoch. Process. Appl. 121(11): 2692–2710. doi:10.1016/j.spa.2011.07.001.
   Web of Science ®Google Scholar
• Bao, J., Wang, F.-Y., Yuan, C. (2013). Bismut formulae and applications for functional SPDEs. Bull. Sci. Math. 137(4): 509–522. doi:10.1016/j.bulsci.2012.11.005.
   Web of Science ®Google Scholar
• Bass, R. F., Levin, D. A. (2002). Harnack inequalities for jump processes. Potential Anal. 17(4): 375–388. doi:10.1023/A:1016378210944.
   Web of Science ®Google Scholar
• Huang, X., Zhang, S.-Q. (2019). Mild solutions and Harnack inequality for functional stochastic partial differential equations with Dini drift. J. Theor. Probab. 32(1): 303–329. doi:10.1007/s10959-018-0830-4.
   Web of Science ®Google Scholar
• Shao, J., Wang, F. Y., Yuan, C. (2012). Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients. Electron. J. Probab. 17: 1–18.
   Web of Science ®Google Scholar
• Wang, F.-Y., Wang, J. (2014). Harnack inequalities for stochastic equations driven by Lévy noise. J. Math. Anal. Appl. 410(1): 513–523. doi:10.1016/j.jmaa.2013.08.013.
   Web of Science ®Google Scholar
• Wang, L., Zhang, X. (2015). Harnack inequalities for SDEs driven by cylindrical α-stable processes. Potential Anal. 42(3): 657–669. doi:10.1007/s11118-014-9451-4.
   Web of Science ®Google Scholar
• Zhang, X. (2013). Derivative formulas and gradient estimates for SDEs driven by α-stable processes. Stoch. Process. Appl. 123(4): 1213–1228. doi:10.1016/j.spa.2012.11.012.
   Web of Science ®Google Scholar
• Fan, X. (2013). Harnack inequality and derivative formula for SDE driven by fractional Brownian motion. Sci. China Math. 56(3): 515–524. doi:10.1007/s11425-013-4569-1.
   Web of Science ®Google Scholar
• Fan, X.-L. (2014). Harnack-Type Inequalities and Applications for SDE Driven by Fractional Brownian Motion. Stochastic Analysis and Applications. 32(4):602–618. doi:10.1080/07362994.2014.907745.
   Web of Science ®Google Scholar
• Li, Z. (2015). Shift Harnack inequality and integration by parts formula for functional SDEs driven by fractional Brownian motion. Proc. Amer. Math. Soc. 144(6): 2651–2659. doi:10.1090/proc/12915.
   Web of Science ®Google Scholar
• Yan, L. T., Yin, X. W. (2018). Harnack inequality and derivative formula for stochastic heat equation with fractional noise. Electron. Comm. Probab. 23(35): 1–11.
   Google Scholar
• Yan, L., Yin, X. (2018). Bismut formula for a stochastic heat equation with fractional noise. Stat. Probab. Lett. 137: 165–172. doi:10.1016/j.spl.2018.01.018.
   Web of Science ®Google Scholar
• Ei Yin, X., Un Shen, G., Ng Gao, Z. (2020). Harnack inequality for stochastic heat equation driven by fractional noise with Hurst index H¿ 1/2. J. Math. Inequ. (4): 1113–1122. doi:10.7153/jmi-2020-14-72.
   Web of Science ®Google Scholar
• Deng, C. S., Huang, X. (2022). Harnack inequalities for functional SDEs driven by subordinate Brownian motions. Potential Anal. 56: 213–226. doi:10.1007/s11118-020-09882-0.
   Google Scholar
• Deng, C. S., Schilling, R. (2017). Harnack inequalities for SDEs driven by time-changed fractional Brownian motions. Electron. J. Probab. 22: 1–23.
   Web of Science ®Google Scholar
• Li, Z., Yan, L. (2018). Harnack inequalities for SDEs driven by subordinator fractional Brownian motion. Stat. Probab. Lett. 134: 45–53. doi:10.1016/j.spl.2017.10.015.
   Web of Science ®Google Scholar
• Deng, C.-S. (2014). Harnack inequalities for SDEs driven by subordinate Brownian motions. J. Math. Anal. Appl. 417(2): 970–978. doi:10.1016/j.jmaa.2014.03.082.
   Web of Science ®Google Scholar
• Rao, M., Song, R., Vondraček, Z. (2006). Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25(1): 1–27. doi:10.1007/s11118-005-9003-z.
   Web of Science ®Google Scholar
• Mimica, A., Kim, P. (2012). Harnack inequalities for subordinate Brownian motions. Electr. J. Probab. 17: 1–23.
   Web of Science ®Google Scholar
• Alòs, E., Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75(3): 129–152. doi:10.1080/1045112031000078917.
   Google Scholar
• Benassi, A., Roux, D., Jaffard, S. (1997). Elliptic gaussian random processes. Rev. Mat. Iberoam. 13(1): 19–90. doi:10.4171/rmi/217.
   Web of Science ®Google Scholar
• Hult, H. (2003). Approximating some Volterra type stochastic integrals with applications to parameter estimation. Stoch. Process. Appl. 105(1): 1–32. doi:10.1016/S0304-4149(02)00250-8.
   Web of Science ®Google Scholar
• Tudor, C. A. (2008). Analysis of the Rosenblatt process. ESAIM: PS. 12: 230–257. doi:10.1051/ps:2007037.
   Google Scholar
• Mishura, Y., Shevchenko, G., Shklyar, S. (2020). Gaussian processes with Volterra kernels. In Silvestrov, S., Malyarenko, A., and Rancic, M., eds. Stochastic Processes, Statistical Methods and Engineering Mathematics. London: Springer.
   Google Scholar
• Research, W., (2008). Kummer confluent hypergeometric function1F1. In The Mathematical Functions Site, 2019–2030. Available at https://www.functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1.
   Google Scholar
• Nunno, G. D., Mishura, Y., Ralchenko, K. (2020). Stochastic differential equations driven by additive Volterra-Lévy and Volterra-Gaussian noises. In: Malyarenko, A., Ni, Y., Rančić, M., Silvestrov, S., Stochastic Processes, Statistical Methods, and Engineering Mathematics. SPAS 2019. Springer Proceedings in Mathematics and Statistics, Vol 408. Cham: Springer. 10.1007/978-3-301-17820-7_14.
   Google Scholar
• Csiszár, I., Körne, J. (1981). Information Theory: Coding Theorems for Discrete Memory-less Systems. New York: Academic Press.
   Google Scholar