Startseite Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Large deviation principle for a space-time fractional stochastic heat equation with fractional noise

  • Litan Yan EMAIL logo und Xiuwei Yin
Veröffentlicht/Copyright: 9. Juni 2018
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 21 Heft 2

Abstract

In this paper, we consider the large deviation principle for a class of space-time fractional stochastic heat equation

∂tβuε(t,x)=−ν(−Δ)α2uε(t,x)+It1−βf(uε(t,x))+εIt1−β[W˙H(t,x)],

where ẆH is a fractional white noise, ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂tβ is the Caputo fractional integration operator, and −(−Δ)α2 is the fractional power of Laplacian. Our proof is based on the weak convergence approach.

MSC 2010: Primary 60F10; Secondary 26A33, 60H15, 60G22
Key Words and Phrases: large deviation; space-time fractional stochastic heat equation; fractional-white noise; weak convergence method

Acknowledgements

The Project-sponsored by NSFC (11571071), Innovation Program of Shanghai Municipal Education Commission(12ZZ063) and The Fundamental Research Funds for the Central Universities (17D310403).

References

[1] R.M. Balan, D. Conus, Intermittency for the wave and heat equations with fractional noise in time. Ann. Probab. 44, No 2 (2016), 1488–1534.10.1214/15-AOP1005Suche in Google Scholar

[2] R.M. Balan, C.A. Tudor, The stochastic heat equation with fractional-colored noise: existence of the solution. LatinAmer. J. Probab. Math. Stat. 4 (2008), 57–87; Latin Amer. J. Probab. Math. Stat. 6 (2009), 343–347 (Erratum).Suche in Google Scholar

[3] R.M. Balan, C.A. Tudor, Stochastic heat equation with multiplicative fractional-colored noise. J. Theor. Probab. 23, No 3 (2010), 834–870.10.1007/s10959-009-0237-3Suche in Google Scholar

[4] V. Bally, A. Millet, M. Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23, No 1 (1995), 178–222.10.1214/aop/1176988383Suche in Google Scholar

[5] K. Bogdan, T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient. Commun. Math. Phys. 271, (2007), 179–198.10.1007/s00220-006-0178-ySuche in Google Scholar

[6] A. Budhiraja, P. Dupuis, V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab. 36, No 4 (2008), 1390–1420.10.21236/ADA476159Suche in Google Scholar

[7] M. Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, Geophys. J. R. Astron. Soc. 13, (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSuche in Google Scholar

[8] S. Cerrai, M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab. 32, No 1(B) (2004), 1100–1139.10.1016/j.anihpb.2004.03.001Suche in Google Scholar

[9] L. Chen, Nonlinear stochastic time-fractional diffusion equations on ℝ: Moments, Hölder regularity and intermittency. Trans. Amer. Math. Soc. 369, No 12 (2017), 8497–8535.10.1090/tran/6951Suche in Google Scholar

[10] L. Chen, G. Hu, Y. Hu, J. Huang, Space-time fractional diffusions in Gaussian noisy enviroment. Stochastics89, No 1 (2017), 171–206.10.1080/17442508.2016.1146282Suche in Google Scholar

[11] L. Chen, K. Kim, On comparision principle and strict positivity of the solutions to the nonlinear stochastic fractonal heat equation. Ann. Inst. Henri Poincaré Probab. Statist. 89, No 1 (2017), 358–388.Suche in Google Scholar

[12] W. Chen, Y. Liang, X. Hei, Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1250–1261; 10.1515/fca-2016-0064; https://www.degruyter.com/view/j/fca.2016.19.issue-5/issue-files/fca.2016.19.issue-5.xml.Suche in Google Scholar

[13] Z.Q. Chen, K.H. Kim, P. Kim, Fractional time stochastic partial differential equations. Stochastic Process Appl. 125, No 4 (2015), 1470–1499.10.1016/j.spa.2014.11.005Suche in Google Scholar

[14] R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s. Electron. J. Probab. 4, No 6 (1999), 1–29.10.1214/EJP.v4-43Suche in Google Scholar

[15] R.C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart, Y. Xiao, A Minicourse on Stochastic Partial Differential Equations. Springer-Verlag, Berlin (2009).10.1007/978-3-540-85994-9Suche in Google Scholar

[16] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations. Wiley (1997).10.1002/9781118165904Suche in Google Scholar

[17] T.E. Mellali, M. Mellouk, Large deviations for a fractional stochastic heat equation in spatial dimension ℝd driven by a spatially correlated noise. Stoch. Dyn. 16, No 1 (2016), # 1650001.10.1142/S0219493716500015Suche in Google Scholar

[18] M. Foondun, W. Liu, M. Omaba, Moment bounds for a class of fractional stochastic heat equation. Ann. Probab. 45, No 4 (2016), 2131–2152.10.1214/16-AOP1108Suche in Google Scholar

[19] M. Foodun, J.B. Mijena, E. Nane, Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1527–1553; 10.1515/fca-2016-0079; https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml.Suche in Google Scholar

[20] M. Foondun, E. Nane, Asymptotic properties of some space-time fractional stochastic equations. Math. Z. 287, No 1-2 (2017), 493–519.10.1007/s00209-016-1834-3Suche in Google Scholar

[21] H. Holden, B. Øsendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations. Springer, Berlin (2010).10.1007/978-0-387-89488-1Suche in Google Scholar

[22] G. Hu, Y. Hu, Fractional diffusion in Gaussian noisy environment. Mathematics3, No 2 (2015), 131–152.10.3390/math3020131Suche in Google Scholar

[23] Y. Hu, D. Nualart, Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields143, No 1-2 (2009), 285–328.10.1007/s00440-007-0127-5Suche in Google Scholar

[24] Y. Hu, D. Nualart, J. Song, Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39, No 1 (2011), 291–326.10.1214/10-AOP547Suche in Google Scholar

[25] D. Khoshnevisan, Analysis of Stochastic Partial Differential Equations. American Mathematical Society (2014).10.1090/cbms/119Suche in Google Scholar

[26] Y. Li, Y. Xie, X. Zhang, Large deviation principle for stochastic heat equation with memory. Discrete Contin. Dyn. Syst. 35, No 11 (2015), 5221–5237.10.3934/dcds.2015.35.5221Suche in Google Scholar

[27] W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction. Springer, New York (2015).10.1007/978-3-319-22354-4Suche in Google Scholar

[28] W. Liu, M. Röcknerb, X.C. Zhu, Large deviation principles for the stochastic quasi-geostrophic equations, Stochastic Process. Appl. 123, No 8 (2013), 3299–3327.10.1016/j.spa.2013.03.020Suche in Google Scholar

[29] D. Márquez-Carreras, M. Sarrá, Large deviation principle for a stochastic heat equation with spatially correlated noise, Electron. J. Probab. 8 (2003), 1–39.10.1214/EJP.v8-146Suche in Google Scholar

[30] A.L. Mehaute, T. Machado, J.C. Trigeassou, J. Sabatier, Fractional Differential and its Applications, FDA’04 (Proc. of the First IFAC Workshop), Vol. 2004-1, International Federation of Autromatic Control, ENSEIRB, Bordeaux, France, July 19-21, 2004.Suche in Google Scholar

[31] J.B. Mijena, E. Nane, Space-time fractional stochastic partial differential equations. Stochastic Process. Appl. 125, No 9 (2015), 3301–3326.10.1016/j.spa.2015.04.008Suche in Google Scholar

[32] J.B. Mijena, E. Nane, Intermittency and time fractional stochastic partial differential equation. Potential Anal. 44, No 2 (2016), 295–312.10.1007/s11118-015-9512-3Suche in Google Scholar

[33] J.B. Mijena, E. Nane, Intermittency fronts for space-time fractional stochastic partial differential equations in (d+1) dimensions. Stochastic Process. Appl. 127, No 4 (2017), 1354–1374.10.1016/j.spa.2016.08.002Suche in Google Scholar

[34] D. Nualart, Malliavin Calculus and Related Topics. Springer-Verlag (2006).Suche in Google Scholar

[35] V. Ortiz-López, M. Sanz-Solé, A Laplace principle for a stochastic wave equation in spatial dimention three. In: Stochastic Analysis, 31–49, Springer-Verlag (2010).Suche in Google Scholar

[36] M. Röcknerb, T.S. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles. Potential Anal. 26, No 3 (2007), 255–279.10.1007/s11118-006-9035-zSuche in Google Scholar

[37] W.R. Schneider, Completely monotone generalized Mittag-Leffler functions. Exposition. Math. 14, No 1 (1996), 3–16.Suche in Google Scholar

[38] T. Simon, Comparing Fréchet and positive stable laws. Electron. J. Probab. 19, No 6 (2014), 1–25.10.1214/EJP.v19-3058Suche in Google Scholar

[39] J. Walsh, An Introduction to Stochastic Partial Differential Equations. École ďété de Probabilités St Flour XIV, Lect. Notes Math, 1180, Springer-Verlag (1986).Suche in Google Scholar

[40] F. Wang, Harnack Inequalities for Stochastic Partial Differential Equations. Springer, Berlin (2014).10.1007/978-1-4614-7934-5Suche in Google Scholar

[41] T.S. Zhang, On small time asymptotics of diffusions on Hilbert spaces. Ann. Probab. 28, No 2 (2000), 537–557.10.1214/aop/1019160252Suche in Google Scholar

[42] T.S. Zhang, Large deviations for stochastic nonlinear beam equations. J. Funct. Anal. 248, No 1 (2007), 175-201.10.1016/j.jfa.2007.03.029Suche in Google Scholar

Received: 2017-1-9
Published Online: 2018-6-9
Published in Print: 2018-4-25

© 2018 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
https://doi.org/10.1515/fca-2018-0026
Schlagwörter für diesen Artikel
large deviation; space-time fractional stochastic heat equation; fractional-white noise; weak convergence method

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2018-0026/pdf?lang=de
Button zum nach oben scrollen