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 & 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.  Google Scholar

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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.  Google Scholar

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I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka Journal of Mathematics, 49 (2012), 421–447.  Google Scholar

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T. Seidman, How violent are fast controls, Mathematics of Control Signals Systems, 1 (1988), 89–95. doi: 10.1007/BF02551238.  Google Scholar

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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.  Google Scholar

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Y. Wang, Hypercontractivity and applications for stochastic Hamiltonian systems, J. Funct. Anal., 272 (2017), 5360–5383. doi: 10.1016/j.jfa.2017.03.015.  Google Scholar

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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.  Google Scholar

[25]

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[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.  Google Scholar

[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.  Google Scholar

[28]

X. C. Zhang, Stochastic hamiltonian flows with singular coefficients, Sci China Math, 61 (2018), 1353–1384. doi: 10.1007/s11425-017-9127-0.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. Bahlali, Flows of homeomorphisms of stochastic differential equations with measurable drift, Stochastic Rep., 67 (1999), 53–82. doi: 10.1080/17442509908834203.  Google Scholar

[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.  Google Scholar

[3]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981.  Google Scholar

[8]

Z.-Q Chen and X. C. Zhang, Propagation of regularity in $L^p$-spaces for Kolmogorov type hypoelliptic operators, arXiv: 1706.02181. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

X. Huang, Strong solutions for functional SDEs with singular drift, Stoch. Dyn., 18 (2018). doi: 10.1142/S0219493718500156.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka Journal of Mathematics, 49 (2012), 421–447.  Google Scholar

[17]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964.  Google Scholar

[18]

T. Seidman, How violent are fast controls, Mathematics of Control Signals Systems, 1 (1988), 89–95. doi: 10.1007/BF02551238.  Google Scholar

[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.  Google Scholar

[20]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

[22]

Y. Wang, Harnack Inequality and Applications for Stochastic Partial Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-7934-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

X. C. Zhang, Stochastic hamiltonian flows with singular coefficients, Sci China Math, 61 (2018), 1353–1384. doi: 10.1007/s11425-017-9127-0.  Google Scholar

[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.  Google Scholar

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