SPDEs with Colored Gaussian Noise: A Survey

Abstract

This note reviews and discusses some recent results on linear stochastic partial differential equations with multiplicative Gaussian noise colored in time.

Keywords

Stochastic partial differential equations / Malliavin Calculus / Large deviation / Lyapunov exponent

References

[1] Bass R, Chen X, Rosen J. Large deviations for Riesz potentials of additive processes. Ann. Inst. Henri Poincaré: Probab. et Stat.. 2009, 45 626-666

[2] Balan RM, Conus D. Intermittency for the wave and heat equations with fractional noise in time. Ann. Probab.. 2016, 44 2 1488-1534

[3] Balan RM, Conus D. A note on intermittency for the fractional heat equation. Stat. Probab. Lett.. 2014, 95 6-14

[4] Balan RM, Jolis M, Quer-Sardanyons L. SPDEs with affine multiplicative fractional noise in space with index $\frac{1}{4}<H<\frac{1}{2}$. Electron. J. Probab.. 2015, 20 54 1-36

[5] Balan, R.M., Quer-Sardanyons, L., Song, J.: Hölder continuity for the parabolic Anderson model with space-time homogeneous Gaussian noise. arXiv

[6] Balan, R.M., Song, J.: Second order Lyapunov exponents for parabolic and hyperbolic Anderson models. arXiv

[7] Balan RM, Song J. Hyperbolic Anderson model with space-time homogeneous Gaussian noise. ALEA Lat. Am. J. Probab. Math. Stat.. 2017, 14 2 799-849

[8] Chen, X.: Parabolic Anderson model with a fractional Gaussian noise that is rough in time. Preprint

[9] Chen, X.: Parabolic Anderson model with rough or critical Gaussian noise. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques. To appear

[10] Chen X. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: in Skorodhod regime. Annales de l’Institut Henri Poincare. 2017, 53 819-841

[11] Chen X. Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann. Probab.. 2016, 44 2 1535-1598

[12] Chen X. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs. 2010 Providence: AMS

[13] Chen X, Rosen J. Large deviations and renormalization for Riesz potentials of stable intersection measures. Stoch. Process. Their Appl.. 2010, 120 1837-1878

[14] Chen, X., Hu, Y., Song, J.: Feynman–Kac formula for fractional heat equation driven by a fractional white noise. arXiv (2014)

[15] Chen, X., Hu, Y., Song, J., Song, X.: Temporal asymptotics for fractional parabolic Anderson model. Electron. J. Probab. 23, Paper No. 14, 39 pp (2018)

[16] Chen X, Hu Y, Song J, Xing F. Exponential asymptotics for time-space Hamiltonians. Annales de l’Institut Henri Poincaré. 2015, 51 1529-1561

[17] Dalang, R.C.: Extending martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electr. J. Probab. 4(6) (1999)

[18] Dalang RC, Mueller C, Tribe R. A Feynman–Kac-type formula for the determinisitic and stochastic wave equations and other p.d.e’s. Trans. Am. Math. Soc.. 2008, 360 4681-4703

[19] Hu Y, Huang J, Le K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann. Probab.. 2017, 45 4561-4616

[20] Hu, Y., Lê, K.: Joint Hölder continuity of parabolic Anderson model. arXiv

[21] Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions. 1992 Cambridge: Cambridge University Press

[22] Folland GB. Introduction to Partial Differential Equations. 1995 2 Princeton: Princeton University Press

[23] Gubinelli M, Imkeller P, Perkowski N. Paracontrolled distributions and singular PDEs. Forum Math. Pi. 2015, 3 e6 75

[24] Gubinelli M. Controlling rough paths. J. Funct. Anal.. 2004, 216 1 86-140

[25] Hairer M. A theory of regularity structures. Invent. Math.. 2014, 198 2 269-504

[26] Huang, J., Lê, K., Nualart, D.: Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise. Stoch. Partial Differ. Equ. To appear

[27] Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab.. 2015, 20 55 50

[28] Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields. 2009, 143 285-328

[29] Hu Y, Nualart D, Song J. Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab.. 2011, 39 1 291-326

[30] Jolis M. The Wiener integral with respect to second order processes with stationary increments. J. Math. Anal. Appl.. 2010, 366 607-620

[31] Le Gall, J.F.: Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités, XXVIII. Lecture Notes in Math. vol. 1583, pp. 172–180. Springer, Berlin (1994)

[32] Lyons TJ. Differential equations driven by rough signals. Rev. Mat. Iberoamericana. 1998, 14 2 215-310

[33] Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. In: Probability and Its Applications (New York), Springer-Verlag, Berlin (2006)

[34] Peszat S, Zabczyk J. Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields. 2000, 116 3 421-443

[35] Prévôt C, Röckner M. A Concise Course on Stochastic Partial Differential Equations. 2007 Berlin: Springer

[36] Rozowskii B. Stochastic Evolution Systems, Mathematics and Its Applications. 1990 Dordrecht: Kluwer

[37] Song J. On a class of stochastic partial differential equations. Stoch. Process. Their Appl.. 2017, 127 1 37-79

[38] Song J. Asymptotic behavior of the solution of heat equation driven by fractional white noise. Stat. Probab. Lett.. 2012, 82 614-620

[39] Walsh, J.B.: An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV-1984, volume 1180 of Lecture Notes in Math., pp. 265–439. Springer, Berlin (1986)

[40] Young LC. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math.. 1936, 67 1 251-282