Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Mohammud Foondun and Davar Khoshnevisan

Full-text: Open access

Abstract

Consider a stochastic heat equation tu=κxx2u+σ(u) for a space–time white noise and a constant κ>0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities

lim sup t→∞t−1sup xRln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup xR|ut(x)|2)

are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process xut(x) are highly concentrated for infinitely-many large values of t. In the special case of the parabolic Anderson model – where σ(u)=λu for some λ>0 – this “peaking” is a way to make precise the notion of physical intermittency.

Résumé

Nous considérons l’équation de la chaleur stochastique tu=κ∂xx2u+σ(u) avec un bruit blanc spatio-temporel et une constante κ>0. Sous des conditions adéquates sur la condition initiale u0 et sur σ, nous montrons que les quantités

lim sup t→∞t−1sup xRln E(|ut(x)|2) et lim sup t→∞t−1ln E(sup xR|ut(x)|2)

sont égales. Par ailleurs, nous les bornons inférieurement et supérieurement par des constantes strictement positives et finies dépendant explicitement de 1/κ. Nos démonstrations reposent sur la preuve quantitative de la forte concentration des pics du processus xut(x) pour de grandes valeurs de t infiniment nombreuses. Dans le cas particulier du modèle d’Anderson parabolique-où σ(u)=λu pour un λ>0 – ce phénomène de pics est une façon de préciser la notion physique d’intermittence.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 895-907.

Dates
First available in Project Euclid: 4 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1288878328

Digital Object Identifier
doi:10.1214/09-AIHP328

Mathematical Reviews number (MathSciNet)
MR2744876

Zentralblatt MATH identifier
1210.35305

Subjects
Primary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Stochastic heat equation Intermittency

Citation

Foondun, Mohammud; Khoshnevisan, Davar. On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 895--907. doi:10.1214/09-AIHP328. https://projecteuclid.org/euclid.aihp/1288878328


Export citation

References

  • [1] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman–Kac formula and intermittence. J. Statist. Physics 78 (1995) 1377–1402.
  • [2] D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probab. 1 (1973) 19–42.
  • [3] R. A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. In Memoires of the AMS 108. Amer. Math. Soc., Rhode Island, 1994.
  • [4] G. Choquet and J. Deny. Sur l’équation de convolution μ=μσ. C. R. Acad. Sci. Paris 250 (1960) 799–801.
  • [5] R. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao. A Minicourse on Stochastic Partial Differential Equations. D. Khoshnevisan and F. Rassoul-Agha (Eds). Lecture Notes in Mathematics 1962. Springer, Berlin, 2009.
  • [6] R. C. Dalang and C. Mueller. Some non-linear s.p.d.e.’s that are second order in time. Electron. J. Probab. 8 (2003). Paper no. 1, 1–21 (electronic).
  • [7] C. Donati-Martin and É. Pardoux. White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 (1993) 1–24.
  • [8] M. Foondun and D. Khoshnevisan. Intermittency and nonlinear parabolic stochastic partial differential equations. Preprint, 2008.
  • [9] M. Foondun, D. Khoshnevisan and E. Nualart. A local time correspondence for stochastic partial differential equations. Preprint, 2008.
  • [10] I. Gyöngy and D. Nualart. On the stochastic Burgers’ equation in the real line. Ann. Probab. 27 (1999) 782–802.
  • [11] M. Kardar. Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B 290 (1987) 582–602.
  • [12] M. Kardar, G. Parisi and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889–892.
  • [13] J. Krug and H. Spohn. Kinetic roughening of growing surfaces. In Solids Far from Equilibrium: Growth, Morphology, and Defects 479–582. C. Godrèche (Ed.). Cambridge Univ. Press, Cambridge, 1991.
  • [14] C. Mueller. On the support of solutions to the heat equation with noise. Stochastics and Stoch. Reports 37 (1991) 225–245.
  • [15] C. Mueller and E. A. Perkins. The compact support property for solutions to the heat equation with noise. Probab. Theory Related Fields 93 (1992) 325–358.
  • [16] T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 (1994) 415–437.
  • [17] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour XIV, 1984 265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986.