## Electronic Journal of Probability

### Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

#### Abstract

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 55, 50 pp.

Dates
Accepted: 23 May 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067161

Digital Object Identifier
doi:10.1214/EJP.v20-3316

Mathematical Reviews number (MathSciNet)
MR3354615

Zentralblatt MATH identifier
1322.60113

Rights

#### Citation

Hu, Yaozhong; Huang, Jingyu; Nualart, David; Tindel, Samy. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20 (2015), paper no. 55, 50 pp. doi:10.1214/EJP.v20-3316. https://projecteuclid.org/euclid.ejp/1465067161

#### References

• Alberts, Tom; Khanin, Konstantin; Quastel, Jeremy. The continuum directed random polymer. J. Stat. Phys. 154 (2014), no. 1-2, 305–326.
• Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël. Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011. xvi+523 pp. ISBN: 978-3-642-16829-1
• R. Balan, D. Conus: Intermittency for the wave and heat equations with fractional noise in time. textitArxiv Preprint (2013).
• Balan, Raluca M.; Tudor, Ciprian A. The stochastic heat equation with fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 57–87.
• Biagini, Francesca; Hu, Yaozhong; Øksendal, Bernt; Zhang, Tusheng. Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications (New York). Springer-Verlag London, Ltd., London, 2008. xii+329 pp. ISBN: 978-1-85233-996-8
• Bertini, Lorenzo; Cancrini, Nicoletta. The stochastic heat equation: Feynman-Kac formula and intermittence. J. Statist. Phys. 78 (1995), no. 5-6, 1377–1401.
• Carmona, Philippe; Hu, Yueyun. Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 217–229.
• Carmona, René A.; Molchanov, S. A. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994), no. 518, viii+125 pp.
• Carmona, R. A.; Molchanov, S. A. Stationary parabolic Anderson model and intermittency. Probab. Theory Related Fields 102 (1995), no. 4, 433–453.
• Caruana, Michael; Friz, Peter K.; Oberhauser, Harald. A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 1, 27–46.
• X.Chen, Y. Hu, J. Song, F. Xing : Exponential asymptotics for time-space Hamiltonians. Preprint.
• Conus, Daniel; Khoshnevisan, Davar. On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Related Fields 152 (2012), no. 3-4, 681–701.
• Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan. On the chaotic character of the stochastic heat equation, II. Probab. Theory Related Fields 156 (2013), no. 3-4, 483–533.
• Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar. On the chaotic character of the stochastic heat equation, before the onset of intermitttency. [On the chaotic character of the stochastic heat equation, before the onset of intermittency] Ann. Probab. 41 (2013), no. 3B, 2225–2260.
• Dalang, Robert C. The stochastic wave equation. A minicourse on stochastic partial differential equations, 39–71, Lecture Notes in Math., 1962, Springer, Berlin, 2009.
• Dalang, Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. (electronic).
• Deya, A.; Gubinelli, M.; Tindel, S. Non-linear rough heat equations. Probab. Theory Related Fields 153 (2012), no. 1-2, 97–147.
• Foondun, Mohammud; Khoshnevisan, Davar. Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 (2009), no. 21, 548–568.
• A.M. Garcia, E. Rodemich, H. Rumsey Jr. : A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Math. J. 20, (1978), 565–578.
• Y. Gu, G. Bal: A Note on Central Limit Theorem for Heat Equation with Large, Highly Oscillatory, Random Potential. textitArxiv preprint (2013).
• M. Gubinelli, P. Imkeller, N. Perkowski: Paracontrolled distributions and singular PDEs. textitArxiv preprint (2013).
• Gubinelli, Massimiliano; Tindel, Samy. Rough evolution equations. Ann. Probab. 38 (2010), no. 1, 1–75.
• Hairer, Martin. Solving the KPZ equation. Ann. of Math. (2) 178 (2013), no. 2, 559–664.
• M. Hairer, E. Pardoux, A. Piatnitski: Random homogenization of a highly oscillatory singular potential. Arxiv preprint, 2013.
• Hu, Yaozhong; Lu, Fei; Nualart, David. Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter $H$ smaller that $1/2$. Ann. Probab. 40 (2012), no. 3, 1041–1068.
• Hu, Yaozhong. Chaos expansion of heat equations with white noise potentials. Potential Anal. 16 (2002), no. 1, 45–66.
• Hu, Yaozhong; Nualart, David. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 (2009), no. 1-2, 285–328.
• Iftimie, Bogdan; Pardoux, Etienne; Piatnitski, Andrey. Homogenization of a singular random one-dimensional PDE. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 3, 519–543.
• Hu, Yaozhong; Nualart, David; Song, Jian. Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39 (2011), no. 1, 291–326.
• Kloeden, P. E.; Neuenkirch, A. The pathwise convergence of approximation schemes for stochastic differential equations. LMS J. Comput. Math. 10 (2007), 235–253.
• KÃ¶nig, Wolfgang; Lacoin, Hubert; Mörters, Peter; Sidorova, Nadia. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009), no. 1, 347–392.
• Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6
• Lacoin, Hubert. Influence of spatial correlation for directed polymers. Ann. Probab. 39 (2011), no. 1, 139–175.
• Le Gall, Jean-François. Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités, XXVIII, 172–180, Lecture Notes in Math., 1583, Springer, Berlin, 1994.
• Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533–597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001.
• Mémin, Jean; Mishura, Yulia; Valkeila, Esko. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51 (2001), no. 2, 197–206.
• Molchanov, Stanislav A. Localization and intermittency: new results. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1091–1103, Math. Soc. Japan, Tokyo, 1991.
• Muirhead, Robb J. Aspects of multivariate statistical theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1982. xix+673 pp. ISBN: 0-471-09442-0
• Nualart, David. Malliavin calculus and its applications. CBMS Regional Conference Series in Mathematics, 110. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2009. viii+85 pp. ISBN: 978-0-8218-4779-4
• Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5.
• Peszat, Szymon; Zabczyk, Jerzy. Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 (1997), no. 2, 187–204.
• Rovira, Carles; Tindel, Samy. On the Brownian-directed polymer in a Gaussian random environment. J. Funct. Anal. 222 (2005), no. 1, 178–201.
• Russo, Francesco; Vallois, Pierre. Elements of stochastic calculus via regularization. Séminaire de Probabilités XL, 147–185, Lecture Notes in Math., 1899, Springer, Berlin, 2007.
• Triebel, Hans. Theory of function spaces. III. Monographs in Mathematics, 100. Birkhauser Verlag, Basel, 2006. xii+426 pp. ISBN: 978-3-7643-7581-2; 3-7643-7581-7.
• Xiao, Yimin. Sample path properties of anisotropic Gaussian random fields. A minicourse on stochastic partial differential equations, 145–212, Lecture Notes in Math., 1962, Springer, Berlin, 2009.