Small time asymptotics for SPDEs with locally monotone coefficients

Shihu Li; Wei Liu; Yingchao Xie

doi:10.3934/dcdsb.2020127

Article Contents

2020, Volume 25, Issue 12: 4801-4822. Doi: 10.3934/dcdsb.2020127

This issue Previous Article Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences Next Article A new weak solution to an optimal stopping problem

Small time asymptotics for SPDEs with locally monotone coefficients

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

^* Corresponding author: Wei Liu
^* Corresponding author: Wei Liu

Received: August 2019

Early access: April 2020

Published: December 2020

The research of W. Liu is supported by NSFC (No. 11822106, 11831014, 11571147), the research of Y. Xie is supported by NSFC (No. 11771187, 11931004) and PAPD of Jiangsu Higher Education Institutions

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Abstract

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic $ p $-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

Keywords:

Small time asymptotics,
large deviation principle,
stochastic partial differential equations,
locally monotone,
porous medium equation.

Mathematics Subject Classification: Primary: 60H15, 60F10; Secondary: 76S05, 35J92, 35K57.

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References

[1]	H. Abdallah, A Varadhan type estimate on manifolds with time-dependent metrics and constant volume, J. Math. Pures Appl., 99 (2013), 409-418. doi: 10.1016/j.matpur.2012.09.007.
[2]	S. Aida and H. Kawabi, Short time asymptotics of a certain infinite dimensional diffusion process, Stochastic Analysis and Related Topics, VII (Kusadasi, 1998), Progr. Probab., Birkh¨auser Boston, Boston, MA, 48 (1998), 77–124.
[3]	S. Aida and T. S. Zhang, On the small time asymptotics of diffusion processes on path groups, Potential Anal., 16 (2002), 67-78. doi: 10.1023/A:1024868720071.
[4]	T. Ariyoshi and M. Hino, Small-time asymptotic estimates in local Dirichlet spaces, Electron. J. Probab., 10 (2005), 1236-1259. doi: 10.1214/EJP.v10-286.
[5]	M. Avellaneda, D. Boyer-Olson, J. Busca and P. Friz, Application of large deviation methods to the pricing of index options in finance, C. R. Math. Acad. Sci. Paris, 336 (2003), 263-266. doi: 10.1016/S1631-073X(03)00032-3.
[6]	H. Berestycki, J. Busca and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 57 (2004), 1352-1373. doi: 10.1002/cpa.20039.
[7]	Z. Brzézniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep., 61 (1997), 245-295. doi: 10.1080/17442509708834122.
[8]	Z. Brzézniak and S. Peszat, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math., 137 (1999), 261-299. doi: 10.4064/sm-137-3-261-299.
[9]	Z. Brzeźniak, W. Liu and J. H. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310. doi: 10.1016/j.nonrwa.2013.12.005.
[10]	A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61.
[11]	A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362.
[12]	Y. Chen, H. J. Gao and L. L. Fan, Well-posedness and the small time large deviations of the stochastic integrable equation governing short-waves in a long-wave model, Nonlinear Anal. Real World Appl., 29 (2016), 38-57. doi: 10.1016/j.nonrwa.2015.10.009.
[13]	Z.-Q. Chen, S. Z. Fang and T. S. Zhang, Small time asymptotics for Brownian motion with singular drift, Proc. Amer. Math. Soc., 147 (2019), 3567-3578. doi: 10.1090/proc/14511.
[14]	P. L. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math., 45 (1992), 97-120. doi: 10.1002/cpa.3160450105.
[15]	I. Chueshov and A. Millet, Stochastic 2D hydrodynamical type systems: Well posedness and large deviations, Appl. Math. Optim., 61 (2010), 379-420. doi: 10.1007/s00245-009-9091-z.
[16]	E. A. Coayla-Teran, P. M. Dias de Magalhães and J. Ferreira, Existence of optimal controls for SPDE with locally monotone coefficients, International J. Control, (2018). doi: 10.1080/00207179.2018.1508849.
[17]	B. Davis, On the $L^p$-norms of stochastic integrals and other martingales, Duke Math. J., 43 (1976), 697-704. doi: 10.1215/S0012-7094-76-04354-4.
[18]	A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, MA, 1993.
[19]	Z. Dong and R. Zhang, On the small-time asymptotics of 3D stochastic primitive equations, Math. Methods Appl. Sci., 41 (2018), 6336-6357. doi: 10.1002/mma.5142.
[20]	S. Fang and T. S. Zhang, On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts, Probab. Theory Related Fields, 114 (1999), 487-504. doi: 10.1007/s004400050232.
[21]	J. Feng, J.-P. Fouque and R. Kumar, Small-time asymptotics for fast mean-reverting stochastic volatility models, Ann. Appl. Probab., 22 (2012), 1541-1575. doi: 10.1214/11-AAP801.
[22]	M. Forde and A. Jacquier, Small-time asymptotics for an uncorrelated local-stochastic volatility model, Appl. Math. Finance, 18 (2011), 517-535. doi: 10.1080/1350486X.2011.591159.
[23]	M. Forde, A. Jacquier and R. Lee, The small-time smile and term structure of implied volatility under the Heston model, SIAM J. Financial Math., 3 (2012), 690-708. doi: 10.1137/110830241.
[24]	J. Frehse and M. Růžička, Non-homogeneous generalized Newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.
[25]	M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, 260. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4684-0176-9.
[26]	B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023.
[27]	B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.
[28]	B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.
[29]	B. Gess, W. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 268 (2020), In press. doi: 10.1016/j.jde.2020.03.002.
[30]	B. L. Guo, C. X. Guo and J. J. Zhang, Martingale and stationary solutions for stochastic non-Newtonian fluids, Differential Integral Equations, 23 (2010), 303-326.
[31]	M. Hino and K. Matsuura, An integrated version of Varadhan's asymptotics for lower-order perturbations of strong local Dirichlet forms, Potential Anal., 48 (2018), 257-300. doi: 10.1007/s11118-017-9634-x.
[32]	M. Hino and J. A. Ramirez, Small-time Gaussian behaviour of symmetric diffusion semigroup, Ann. Probab., 31 (2003), 1254-1295. doi: 10.1214/aop/1055425779.
[33]	T. Jegaraj, Small time asymptotics for stochastic evolution equations, J. Theoret. Probab., 24 (2011), 756-788. doi: 10.1007/s10959-010-0336-1.
[34]	N. V. Krylov and B. L. Rozovskii, Stochastic evolution equations, Stochastic Differential Equations: Theory and Applications, 1–69, Interdiscip. Math. Sci., Vol. 2, World Sci. Publ., Hackensack, NJ, 2007. doi: 10.1142/9789812770639_0001.
[35]	O. A. Ladyzhenskaya, New equations for the description of the viscous incompressible fluids and solvability in large of the boundary value problems for them, Volume V of Boundary Value Problems of Mathematical Physics, Amer. Math. Soc., Providence, 1970.
[36]	S. H. Li, W. Liu and Y. C. Xie, Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation, Commun. Pure Appl. Anal., 18 (2019), 2491-2510. doi: 10.3934/cpaa.2019113.
[37]	H. Liu and C. F. Sun, On the small time asymptotics of stochastic non-Newtonian fluids, Math. Methods Appl. Sci., 40 (2017), 1139-1152. doi: 10.1002/mma.4041.
[38]	W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ., 9 (2009), 747-770. doi: 10.1007/s00028-009-0032-8.
[39]	W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56. doi: 10.1007/s00245-009-9072-2.
[40]	W. Liu and M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259 (2010), 2902-2922. doi: 10.1016/j.jfa.2010.05.012.
[41]	W. Liu and M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Differential Equations, 254 (2013), 725-755. doi: 10.1016/j.jde.2012.09.014.
[42]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.
[43]	W. Liu, M. Röckner and J. L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives, SIAM J. Math. Anal., 50 (2018), 2588-2607. doi: 10.1137/17M1144593.
[44]	W. Liu, M. Röckner and X.-C. Zhu, Large deviation principles for the stochastic quasi-geostrophic equations, Stochastic Process. Appl., 123 (2013), 3299-3327. doi: 10.1016/j.spa.2013.03.020.
[45]	W. Liu, C. Tao and J. Zhu, Large deviation principle for a class of SPDE with locally monotone coefficients, Sci. China Math., (2020), In press. doi: 10.1007/s11425-018-9440-3.
[46]	T. Ma and R.-C. Zhu, Wong-Zakai approximation and support theorem for SPDEs with locally monotone coefficients, J. Math. Anal. Appl., 469 (2019), 623-660. doi: 10.1016/j.jmaa.2018.09.031.
[47]	J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13. Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.
[48]	E. Pardoux, Equations aux Dérivées Partielles Stochastiques non Linéaires Monotones, Theèse de Doctorat, Université Paris-Sud, 1975.
[49]	J. G. Ren and X. C. Zhang, Freidlin-wentzell's large deviations for stochastic evolution equations, J. Funct. Anal., 254 (2008), 3148-3172. doi: 10.1016/j.jfa.2008.02.010.
[50]	M. Röckner and T. S. Zhang, Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles, J. Differential Equations, 252 (2012), 716-744. doi: 10.1016/j.jde.2011.09.030.
[51]	J. Seidler, Exponential estimates for stochastic convolutions in 2-smooth Banach spaces, Electron. J. Probab., 15 (2010), 1556-1573. doi: 10.1214/EJP.v15-808.
[52]	D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1.
[53]	K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems, London Mathematical Society Lecture Note Series, 223. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511662362.
[54]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995. 532 pp.
[55]	S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 19 (1966), 261-286. doi: 10.1002/cpa.3160190303.
[56]	S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20 (1967), 431-455. doi: 10.1002/cpa.3160200210.
[57]	S. R. S. Varadhan, Diffusion processes in a small time interval, Comm. Pure Appl. Math., 20 (1967), 659-685. doi: 10.1002/cpa.3160200404.
[58]	F.-Y. Wang, Exponential convergence of non-linear monotone SPDEs, Discrete Contin. Dyn. Syst., 35 (2015), 5239-5253. doi: 10.3934/dcds.2015.35.5239.
[59]	J. Xiong and J. L. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise, Bernoulli, 24 (2018), 2842-2874. doi: 10.3150/17-BEJ947.
[60]	T. G. Xu and T. S. Zhang, On the small time asymptotics of the two-dimensional stochastic Navier-Stokes equations, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 1002-1019. doi: 10.1214/08-AIHP192.
[61]	R. Zhang, On the small time asymptotics of scalar stochastic conservation laws, arXiv: 1907.03397.
[62]	T. S. Zhang, On the small time asymptotics of diffusion processes on Hilbert spaces, Ann. Probab., 28 (2000), 537-557. doi: 10.1214/aop/1019160251.
[63]	X. C. Zhang, Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, J. Funct. Anal., 258 (2010), 1361-1425. doi: 10.1016/j.jfa.2009.11.006.
[64]	J. H. Zhu, Z. Brzezniak and W. Liu, Maximal inequalities and exponential estimates for stochastic convolutions driven by Lévy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations, SIAM J. Math. Anal., 51 (2019), 2121-2167. doi: 10.1137/18M1169011.