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 and 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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. 

[20]

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

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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. 

[20]

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

[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.

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