September  2020, 40(9): 5173-5188. doi: 10.3934/dcds.2020224

Limit theorems for additive functionals of path-dependent SDEs

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK

* Corresponding author: Chenggui Yuan

Received  June 2019 Published  June 2020

Fund Project: This work is supported in part by NNSFC (11771326, 11431014, 11831014)

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

Citation: Jianhai Bao, Feng-Yu Wang, Chenggui Yuan. Limit theorems for additive functionals of path-dependent SDEs. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5173-5188. doi: 10.3934/dcds.2020224
References:
[1]

J. H. BaoF.-Y. Wang and C. G. Yuan, Hypercontractivity for functional stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3636-3656.  doi: 10.1016/j.spa.2015.04.001.

[2]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for neutral type SDEs with infinite length of memory, Math. Nach., arXiv: 1805.03431.

[3]

W. BoltA. A. Majewski and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math., 212 (2012), 41-53.  doi: 10.4064/sm212-1-3.

[4]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), Paper No. 98, 23 pp. doi: 10.1214/17-EJP122.

[5]

O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, Ann. Appl. Probab., 24 (2014), 526-552.  doi: 10.1214/13-AAP922.

[6]

P. CattiauxD. Chafai and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382. 

[7]

X. Chen, The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self-normalization, J. Theoret. Probab., 12 (1999), 421-445.  doi: 10.1023/A:1021630228280.

[8]

Y. Derriennic and M. Lin, The central limit theorem for Markov chains started at a point, Probab. Theory Related Fields, 125 (2003), 73-76.  doi: 10.1007/s004400200215.

[9]

W. Doeblin, Sur deux problemes de M. Kolmogoroff concernant les chanes d énombrables, Bull. Soc. Math. France, 66 (1938), 210-220. 

[10]

G. Dos Reis, W. Salkeld and J. Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the Functional Iterated Logarithm Law, arXiv: 1708.04961v3.

[11]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.

[12]

F. Q. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), Paper No. 94, 21 pp. doi: 10.1214/17-EJP104.

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. (2), 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.

[14]

M. HairerJ. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.

[15] P. Hall and C. C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York-London, 1980. 
[16]

C. C. Heyde and D. J. Scott, Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1 (1973), 428-436.  doi: 10.1214/aop/1176996937.

[17]

I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.

[18]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

[19]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.

[20]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusions, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.

[21]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122 (2012), 2155-2184.  doi: 10.1016/j.spa.2012.03.006.

[22]

A. Kulik, Ergodic Behavior of Markov Processes. With Applications to Limit Theorems, De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2018.

[23]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.

[24]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247.  doi: 10.1007/s00440-005-0427-6.

[25]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1964), 211-226.  doi: 10.1007/BF00534910.

[26]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ., 18 (2010), 267-284.  doi: 10.1515/ROSE.2010.015.

[27]

A. Walczuk, Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 62 (2008), 149-159.  doi: 10.2478/v10062-008-0016-0.

[28]

L. M. Wu, Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 121-141.  doi: 10.1016/S0246-0203(99)80008-9.

show all references

References:
[1]

J. H. BaoF.-Y. Wang and C. G. Yuan, Hypercontractivity for functional stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3636-3656.  doi: 10.1016/j.spa.2015.04.001.

[2]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for neutral type SDEs with infinite length of memory, Math. Nach., arXiv: 1805.03431.

[3]

W. BoltA. A. Majewski and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math., 212 (2012), 41-53.  doi: 10.4064/sm212-1-3.

[4]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), Paper No. 98, 23 pp. doi: 10.1214/17-EJP122.

[5]

O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, Ann. Appl. Probab., 24 (2014), 526-552.  doi: 10.1214/13-AAP922.

[6]

P. CattiauxD. Chafai and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382. 

[7]

X. Chen, The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self-normalization, J. Theoret. Probab., 12 (1999), 421-445.  doi: 10.1023/A:1021630228280.

[8]

Y. Derriennic and M. Lin, The central limit theorem for Markov chains started at a point, Probab. Theory Related Fields, 125 (2003), 73-76.  doi: 10.1007/s004400200215.

[9]

W. Doeblin, Sur deux problemes de M. Kolmogoroff concernant les chanes d énombrables, Bull. Soc. Math. France, 66 (1938), 210-220. 

[10]

G. Dos Reis, W. Salkeld and J. Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the Functional Iterated Logarithm Law, arXiv: 1708.04961v3.

[11]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.

[12]

F. Q. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), Paper No. 94, 21 pp. doi: 10.1214/17-EJP104.

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. (2), 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.

[14]

M. HairerJ. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.

[15] P. Hall and C. C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York-London, 1980. 
[16]

C. C. Heyde and D. J. Scott, Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1 (1973), 428-436.  doi: 10.1214/aop/1176996937.

[17]

I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.

[18]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

[19]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.

[20]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusions, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.

[21]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122 (2012), 2155-2184.  doi: 10.1016/j.spa.2012.03.006.

[22]

A. Kulik, Ergodic Behavior of Markov Processes. With Applications to Limit Theorems, De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2018.

[23]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.

[24]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247.  doi: 10.1007/s00440-005-0427-6.

[25]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1964), 211-226.  doi: 10.1007/BF00534910.

[26]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ., 18 (2010), 267-284.  doi: 10.1515/ROSE.2010.015.

[27]

A. Walczuk, Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 62 (2008), 149-159.  doi: 10.2478/v10062-008-0016-0.

[28]

L. M. Wu, Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 121-141.  doi: 10.1016/S0246-0203(99)80008-9.

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