September  2021, 20(9): 3129-3142. doi: 10.3934/cpaa.2021099

Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs

1. 

Mathematic Department, City University of Hong Kong, Hong Kong, China

2. 

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

* Corresponding author

Received  January 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.

Citation: Panpan Ren, Shen Wang. Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3129-3142. doi: 10.3934/cpaa.2021099
References:
[1]

P. A. Baldi, Large deviations and stochastic homogenisation, Ann. Mat. Pura Appl., 151 (1988), 161-177.  doi: 10.1007/BF01762793.  Google Scholar

[2]

A. A. Borovkov and A. A. Mogulskii, Probabilities of large deviations in topological vector space I, Siberian Math. J., 19 (1978), 697-709.   Google Scholar

[3]

A. A. Borovkov and A. A. Mogulskii, Probabilities of large deviations in topological vector space II, Siberian Math. J., 21 (1980), 12-26.   Google Scholar

[4]

J. BaoF. Y. Wang and C. Yuan, Limit theorems for additive functionals of path-dependent SDEs, Discrete Contin. Dyn. Syst., 40 (2020), 5173-5188.  doi: 10.3934/dcds.2020224.  Google Scholar

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X. Chen, The moderate deviations of independent random vectors in a Banach space, Chinese J. Appl. Probab. Statist., 7 (1991), 24-32.   Google Scholar

[6]

P. CattiauxP. Dai Pra and S. Roelly, A constructive approach to a class of ergodic HJB equatons with unbounded and nonsmooth cost, SIAM J. Control Optim., 47 (2008), 2598-2615.  doi: 10.1137/070698634.  Google Scholar

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M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I-IV, Comm. Pure Appl. Math., 28 (1975), 1-47, 279-301; 29(1976), 389-461; 36(1983), 183-212. doi: 10.1002/cpa.3160280102.  Google Scholar

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A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, $2^{nd}$ edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

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F. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), 1-21.  doi: 10.1214/17-EJP104.  Google Scholar

[10]

X. Huang, P. Ren and F. Y. Wang, Distribution Dependent Stochastic Differential Equation, preprint, arXiv: 2012.13656. Google Scholar

[11]

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.  Google Scholar

[12]

I. Kontoyiannis and S. P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann. Appl. Probab., 13 (2003), 304-362.  doi: 10.1214/aoap/1042765670.  Google Scholar

[13]

P. Ren and F. Y. Wang, Donsker-Varadhan Large Deviations for Path-Distribution Dependent SPDEs, preprint, arXiv: 2002.08652. doi: 10.1016/j. jmaa. 2021.125000.  Google Scholar

[14]

M. RöcknerF. Y. Wang and L. Wu, Large deviations for stochastic generalized porous media equations, Stoch. Proc. Appl., 116 (2006), 1677-1689.  doi: 10.1016/j.spa.2006.05.007.  Google Scholar

[15]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979.  Google Scholar

[16]

F. Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex mainfolds, Ann. Probab., 39 (2011), 1449-1467.  doi: 10.1214/10-AOP600.  Google Scholar

[17]

F. 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

[18]

F. Y. Wang, Distribution dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.  Google Scholar

[19]

F. Y. Wang and Y. Zhang, Application of Harnack inequality to long time asymptotics of Markov processes(in Chinese), Sci. Sin. Math., 49 (2019), 505-516.   Google Scholar

[20]

L. Wu, Moderate deviations of dependent random variables related to CLT, Ann. Probab., 23 (1995), 420-445.   Google Scholar

[21]

L. Wu, Uniformly integrable operators and large deviations for Markov processes, J. Funct. Anal., 172 (2000), 301-376.  doi: 10.1006/jfan.1999.3544.  Google Scholar

show all references

References:
[1]

P. A. Baldi, Large deviations and stochastic homogenisation, Ann. Mat. Pura Appl., 151 (1988), 161-177.  doi: 10.1007/BF01762793.  Google Scholar

[2]

A. A. Borovkov and A. A. Mogulskii, Probabilities of large deviations in topological vector space I, Siberian Math. J., 19 (1978), 697-709.   Google Scholar

[3]

A. A. Borovkov and A. A. Mogulskii, Probabilities of large deviations in topological vector space II, Siberian Math. J., 21 (1980), 12-26.   Google Scholar

[4]

J. BaoF. Y. Wang and C. Yuan, Limit theorems for additive functionals of path-dependent SDEs, Discrete Contin. Dyn. Syst., 40 (2020), 5173-5188.  doi: 10.3934/dcds.2020224.  Google Scholar

[5]

X. Chen, The moderate deviations of independent random vectors in a Banach space, Chinese J. Appl. Probab. Statist., 7 (1991), 24-32.   Google Scholar

[6]

P. CattiauxP. Dai Pra and S. Roelly, A constructive approach to a class of ergodic HJB equatons with unbounded and nonsmooth cost, SIAM J. Control Optim., 47 (2008), 2598-2615.  doi: 10.1137/070698634.  Google Scholar

[7]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I-IV, Comm. Pure Appl. Math., 28 (1975), 1-47, 279-301; 29(1976), 389-461; 36(1983), 183-212. doi: 10.1002/cpa.3160280102.  Google Scholar

[8]

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, $2^{nd}$ edition, Springer, New York, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[9]

F. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), 1-21.  doi: 10.1214/17-EJP104.  Google Scholar

[10]

X. Huang, P. Ren and F. Y. Wang, Distribution Dependent Stochastic Differential Equation, preprint, arXiv: 2012.13656. Google Scholar

[11]

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.  Google Scholar

[12]

I. Kontoyiannis and S. P. Meyn, Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann. Appl. Probab., 13 (2003), 304-362.  doi: 10.1214/aoap/1042765670.  Google Scholar

[13]

P. Ren and F. Y. Wang, Donsker-Varadhan Large Deviations for Path-Distribution Dependent SPDEs, preprint, arXiv: 2002.08652. doi: 10.1016/j. jmaa. 2021.125000.  Google Scholar

[14]

M. RöcknerF. Y. Wang and L. Wu, Large deviations for stochastic generalized porous media equations, Stoch. Proc. Appl., 116 (2006), 1677-1689.  doi: 10.1016/j.spa.2006.05.007.  Google Scholar

[15]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979.  Google Scholar

[16]

F. Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex mainfolds, Ann. Probab., 39 (2011), 1449-1467.  doi: 10.1214/10-AOP600.  Google Scholar

[17]

F. 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

[18]

F. Y. Wang, Distribution dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.  Google Scholar

[19]

F. Y. Wang and Y. Zhang, Application of Harnack inequality to long time asymptotics of Markov processes(in Chinese), Sci. Sin. Math., 49 (2019), 505-516.   Google Scholar

[20]

L. Wu, Moderate deviations of dependent random variables related to CLT, Ann. Probab., 23 (1995), 420-445.   Google Scholar

[21]

L. Wu, Uniformly integrable operators and large deviations for Markov processes, J. Funct. Anal., 172 (2000), 301-376.  doi: 10.1006/jfan.1999.3544.  Google Scholar

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