American Institute of Mathematical Sciences

March  2020, 19(3): 1257-1273. doi: 10.3934/cpaa.2020060

Stochastic functional Hamiltonian system with singular coefficients

 1 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 2 Department of Statistics, College of Science, Donghua University, Shanghai 201620, China

* Corresponding author

Received  February 2019 Revised  August 2019 Published  November 2019

By Zvonkin type transforms, the existence and uniqueness of the strong solutions for a class of stochastic functional Hamiltonian systems are obtained, where the drift contains a Hölder-Dini continuous perturbation. Moreover, under some reasonable conditions, the non-explosion of the solution is proved. In addition, as applications, the Harnack and shift Harnack inequalities are derived by method of coupling by change of measure. These inequalities are new even in the case without delay and the shift Harnack inequality is also new even in the non-degenerate functional SDEs with singular drifts.

Citation: Xing Huang, Wujun Lv. Stochastic functional Hamiltonian system with singular coefficients. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1257-1273. doi: 10.3934/cpaa.2020060
References:
 [1] K. Bahlali, Flows of homeomorphisms of stochastic differential equations with measurable drift, Stochastic Rep., 67 (1999), 53–82. doi: 10.1080/17442509908834203. [2] S. Bachmann, Well-posedness and stability for a class of stochastic delay differential equations with singular drift, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500193. [3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434. [4] J. Bao, F.-Y. Wang and C. Yuan, Derivative formula and Harnack inequality for degenerate functionals SDEs, Stoch. Dyn., 13 (2013), 943–951. doi: 10.1142/S021949371250013X. [5] E. Chaudru de Raynal, Weak regularization by stochastic drift: result and counter example, Discrete Cont Dyn-A, 38 (2018), 1269–1291. doi: 10.3934/dcds.2018052. [6] E. Chaudru de Raynal and S. Menozzi, Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result, arXiv: 1710.03620. [7] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. [8] Z.-Q Chen and X. C. Zhang, Propagation of regularity in $L^p$-spaces for Kolmogorov type hypoelliptic operators, arXiv: 1706.02181. [9] E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electron J Probab, 22 (2017), 1–48. doi: 10.1214/17-EJP65. [10] A. Guillin and F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differential Equations, 253 (2012), 20–40. doi: 10.1016/j.jde.2012.03.014. [11] L. Gyöngy and T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J., 51 (2001), 763–783. doi: 10.1023/A:1013764929351. [12] X. Huang, Strong solutions for functional SDEs with singular drift, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500156. [13] X. Huang and F.-Y. Wang, Functional SPDE with multiplicative noise and Dini drift, Ann. Fac. Sci. Toulouse Math., 6 (2017), 519–537. doi: 10.5802/afst.1544. [14] X. Huang and F.-Y. Wang, Degenerate SDEs with singular drift and applications to Heisenberg groups, J. Differential Equations, 265 (2018), 2745–2777. doi: 10.1016/j.jde.2018.04.050. [15] X. Huang and S.-Q. Zhang, Mild solutions and Harnack inequality for functional stochastic partial differential equations with Dini drift, J. Theoret. Probab., 32 (2019), 303–329. doi: 10.1007/s10959-018-0830-4. [16] E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka Journal of Mathematics, 49 (2012), 421–447. [17] M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. [18] T. Seidman, How violent are fast controls, Mathematics of Control Signals Systems, 1 (1988), 89–95. doi: 10.1007/BF02551238. [19] J. Shao, F.-Y. Wang and C. Yuan, Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients, Elect. J. Probab., 17 (2012), 1–18. doi: 10.1214/EJP.v17-2140. [20] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. [21] Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differential Equations, 260 (2016), 2792–2829. doi: 10.1016/j.jde.2015.10.020. [22] Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5. [23] Y. Wang, Hypercontractivity and applications for stochastic Hamiltonian systems, J. Funct. Anal., 272 (2017), 5360–5383. doi: 10.1016/j.jfa.2017.03.015. [24] Y. Wang and X. C. Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl., 99 (2013), 726–740. doi: 10.1016/j.matpur.2012.10.007. [25] Y. Wang and X. C. Zhang, Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient, SIAM J. Math. Anal., 48 (2016), 2189–2226. doi: 10.1137/15M1023671. [26] X. C. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stoch. Proc. Appl., 120 (2010), 1929–1949. doi: 10.1016/j.spa.2010.05.015. [27] X. C. Zhang, Strong solutions of SDEs with singural drift and Sobolev diffusion coefficients, Stoch. Proc. Appl., 115 (2005), 1805–1818. doi: 10.1016/j.spa.2005.06.003. [28] X. C. Zhang, Stochastic hamiltonian flows with singular coefficients, Sci China Math, 61 (2018), 1353–1384. doi: 10.1007/s11425-017-9127-0. [29] A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129–149,152.

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References:
 [1] K. Bahlali, Flows of homeomorphisms of stochastic differential equations with measurable drift, Stochastic Rep., 67 (1999), 53–82. doi: 10.1080/17442509908834203. [2] S. Bachmann, Well-posedness and stability for a class of stochastic delay differential equations with singular drift, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500193. [3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434. [4] J. Bao, F.-Y. Wang and C. Yuan, Derivative formula and Harnack inequality for degenerate functionals SDEs, Stoch. Dyn., 13 (2013), 943–951. doi: 10.1142/S021949371250013X. [5] E. Chaudru de Raynal, Weak regularization by stochastic drift: result and counter example, Discrete Cont Dyn-A, 38 (2018), 1269–1291. doi: 10.3934/dcds.2018052. [6] E. Chaudru de Raynal and S. Menozzi, Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result, arXiv: 1710.03620. [7] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. [8] Z.-Q Chen and X. C. Zhang, Propagation of regularity in $L^p$-spaces for Kolmogorov type hypoelliptic operators, arXiv: 1706.02181. [9] E. Fedrizzi, F. Flandoli, E. Priola and J. Vovelle, Regularity of stochastic kinetic equations, Electron J Probab, 22 (2017), 1–48. doi: 10.1214/17-EJP65. [10] A. Guillin and F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differential Equations, 253 (2012), 20–40. doi: 10.1016/j.jde.2012.03.014. [11] L. Gyöngy and T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J., 51 (2001), 763–783. doi: 10.1023/A:1013764929351. [12] X. Huang, Strong solutions for functional SDEs with singular drift, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500156. [13] X. Huang and F.-Y. Wang, Functional SPDE with multiplicative noise and Dini drift, Ann. Fac. Sci. Toulouse Math., 6 (2017), 519–537. doi: 10.5802/afst.1544. [14] X. Huang and F.-Y. Wang, Degenerate SDEs with singular drift and applications to Heisenberg groups, J. Differential Equations, 265 (2018), 2745–2777. doi: 10.1016/j.jde.2018.04.050. [15] X. Huang and S.-Q. Zhang, Mild solutions and Harnack inequality for functional stochastic partial differential equations with Dini drift, J. Theoret. Probab., 32 (2019), 303–329. doi: 10.1007/s10959-018-0830-4. [16] E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka Journal of Mathematics, 49 (2012), 421–447. [17] M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. [18] T. Seidman, How violent are fast controls, Mathematics of Control Signals Systems, 1 (1988), 89–95. doi: 10.1007/BF02551238. [19] J. Shao, F.-Y. Wang and C. Yuan, Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients, Elect. J. Probab., 17 (2012), 1–18. doi: 10.1214/EJP.v17-2140. [20] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. [21] Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differential Equations, 260 (2016), 2792–2829. doi: 10.1016/j.jde.2015.10.020. [22] Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5. [23] Y. Wang, Hypercontractivity and applications for stochastic Hamiltonian systems, J. Funct. Anal., 272 (2017), 5360–5383. doi: 10.1016/j.jfa.2017.03.015. [24] Y. Wang and X. C. Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl., 99 (2013), 726–740. doi: 10.1016/j.matpur.2012.10.007. [25] Y. Wang and X. C. Zhang, Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient, SIAM J. Math. Anal., 48 (2016), 2189–2226. doi: 10.1137/15M1023671. [26] X. C. Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stoch. Proc. Appl., 120 (2010), 1929–1949. doi: 10.1016/j.spa.2010.05.015. [27] X. C. Zhang, Strong solutions of SDEs with singural drift and Sobolev diffusion coefficients, Stoch. Proc. Appl., 115 (2005), 1805–1818. doi: 10.1016/j.spa.2005.06.003. [28] X. C. Zhang, Stochastic hamiltonian flows with singular coefficients, Sci China Math, 61 (2018), 1353–1384. doi: 10.1007/s11425-017-9127-0. [29] A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129–149,152.
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