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Moderate deviations for a stochastic Burgers equation
Volume 6, Issue 2 (2019), pp. 167–193
Rachid Belfadli   Lahcen Boulanba   Mohamed Mellouk  

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https://doi.org/10.15559/19-VMSTA134
Pub. online: 16 May 2019      Type: Research Article      Open accessOpen Access

Received
29 October 2018
Revised
29 March 2019
Accepted
17 April 2019
Published
16 May 2019

Abstract

A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

References

[1] 
Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation. Commun. Math. Phys. 165(2), 211–232 (1994). MR1301846
[2] 
Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998). MR1675051. https://doi.org/10.1214/aop/1022855876
[3] 
Bryc, W.: Large deviations by the asymptotic value method. Diffus. Process. Relat. Probl. Anal. 20, 1004–1030 (1992)
[4] 
Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat., Wroclaw Univ. 20(1), 39–61 (2000). MR1785237
[5] 
Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab., 1390–1420 (2008). MR2435853. https://doi.org/10.1214/07-AOP362
[6] 
Budhiraja, A., Dupuis, P., Ganguly, A., et al.: Moderate deviation principles for stochastic differential equations with jumps. Ann. Probab. 44(3), 1723–1775 (2016). MR3502593. https://doi.org/10.1214/15-AOP1007
[7] 
Burgers, J.M.: The nonlinear diffusion equation. Asymptotic solutions and statistical problems, D. Reidel, Dordrecht-H, Boston (1974)
[8] 
Cardon-Weber, C.: Large deviations for a Burgers-type SPDE. Stoch. Process. Appl. 84(1), 53–70 (1999). MR1720097. https://doi.org/10.1016/S0304-4149(99)00047-2
[9] 
Cardon-Weber, C., Millet, A.: A support theorem for a generalized Burgers SPDE. Potential Anal. 15(4), 361–408 (2001). MR1856154. https://doi.org/10.1023/A:1011857909744
[10] 
Chen, Y., Gao, H.: Well-posedness and large deviations for a class of SPDEs with Lévy noise. J. Differ. Equ. 263(9), 5216–5252 (2017). MR3688413. https://doi.org/10.1016/j.jde.2017.06.016
[11] 
Chenal, F., Millet, A.: Uniform large deviations for parabolic SPDEs and applications. Stoch. Process. Appl. 72(2), 161–186 (1997). MR1486551. https://doi.org/10.1016/S0304-4149(97)00091-4
[12] 
De Acosta, A.: Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Am. Math. Soc. 329(1), 357–375 (1992). MR1046015. https://doi.org/10.2307/2154092
[13] 
Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations, vol. 902. John Wiley & Sons (2011). MR1431744. https://doi.org/10.1002/9781118165904
[14] 
Foondun, M., Setayeshgar, L.: Large deviations for a class of semilinear stochastic partial differential equations. Stat. Probab. Lett. 121, 143–151 (2017). MR3575422. https://doi.org/10.1016/j.spl.2016.10.019
[15] 
Freidlin, M., Wentzell, A.: Random perturbations of dynamical systems. Springer (1984). MR0722136. https://doi.org/10.1007/978-1-4684-0176-9
[16] 
Gao, F.-Q.: Moderate deviations for martingales and mixing random processes. Stoch. Process. Appl. 61(2), 263–275 (1996). MR1386176. https://doi.org/10.1016/0304-4149(95)00078-X
[17] 
Gyöngy, I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73(2), 271–299 (1998). MR1608641. https://doi.org/10.1016/S0304-4149(97)00103-8
[18] 
Hu, S., Li, R., Wang, X.: Central limit theorem and moderate deviations for a class of semilinear SPDEs. arXiv preprint arXiv:1811.05611 (2018)
[19] 
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes vol. 24. Elsevier (2014). MR0892528
[20] 
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, vol. 113. Springer (2012). MR1121940. https://doi.org/10.1007/978-1-4612-0949-2
[21] 
Liming, W.: Moderate deviations of dependent random variables related to CLT. Ann. Probab., 420–445 (1995). MR1330777
[22] 
Morien, P.-L.: On the density for the solution of a Burgers-type SPDE. In: Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 35, pp. 459–482 (1999). Elsevier. MR1702238. https://doi.org/10.1016/S0246-0203(99)00102-8
[23] 
Setayeshgar, L.: Large deviations for a stochastic Burgers equation. Commun. Stoch. Anal. 8, 141–154 (2014). MR3269841. https://doi.org/10.31390/cosa.8.2.01
[24] 
Varadhan, S.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19(3), 261–286 (1966). MR0203230. https://doi.org/10.1002/cpa.3160190303
[25] 
Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’Été de Probabilités de Saint Flour XIV-1984, pp. 265–439. Springer (1986). MR0876085. https://doi.org/10.1007/BFb0074920
[26] 
Wang, R., Zhang, T.: Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Potential Anal. 42(1), 99–113 (2015). MR3297988. https://doi.org/10.1007/s11118-014-9425-6
[27] 
Yang, J., Jiang, Y.: Moderate deviations for fourth-order stochastic heat equations with fractional noises. Stoch. Dyn. 16(06), 1650022 (2016). MR3568726. https://doi.org/10.1142/S0219493716500222
[28] 
Zhang, R., Xiong, J.: Semilinear stochastic partial differential equations: central limit theorem and moderate deviations. arXiv preprint arXiv:1904.00299 (2019)

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Keywords
Stochastic Burgers equation space-time white noise stochastic partial differential equations moderate deviations principle weak convergence method

MSC2010
60F10 60F05 60H15

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