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

Foondun, Mohammud; Khoshnevisan, Davar

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 895-907.

Nous considérons l'équation de la chaleur stochastique ∂tu=κ∂xx2u+σ(u)ẇ avec un bruit blanc spatio-temporel ẇ 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 x∈Rln E(|ut(x)|2) et lim sup t→∞t-1ln E(sup x∈R|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 x↦ut(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.

Consider a stochastic heat equation ∂tu=κ ∂xx2u+σ(u)ẇ for a space-time white noise ẇ 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 x∈Rln El(|ut(x)|2) and lim sup t→∞t-1ln E(sup x∈R|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 x↦ut(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.

Classification : 35R60, 37H10, 60H15, 82B44
Mots clés : stochastic heat equation, intermittency

[1] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman-Kac formula and intermittence. J. Statist. Physics 78 (1995) 1377-1402. | MR | Zbl

[2] D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probab. 1 (1973) 19-42. | MR | Zbl

[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. | MR | Zbl

[4] G. Choquet and J. Deny. Sur l'équation de convolution μ=μ∗σ. C. R. Acad. Sci. Paris 250 (1960) 799-801. | MR | Zbl

[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. | MR

[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). | MR | Zbl

[7] C. Donati-Martin and É. Pardoux. White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 (1993) 1-24. | MR | Zbl

[8] M. Foondun and D. Khoshnevisan. Intermittency and nonlinear parabolic stochastic partial differential equations. Preprint, 2008. | MR | Zbl

[9] M. Foondun, D. Khoshnevisan and E. Nualart. A local time correspondence for stochastic partial differential equations. Preprint, 2008. | MR

[10] I. Gyöngy and D. Nualart. On the stochastic Burgers' equation in the real line. Ann. Probab. 27 (1999) 782-802. | MR | Zbl

[11] M. Kardar. Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B 290 (1987) 582-602. | MR

[12] M. Kardar, G. Parisi and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889-892. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[16] T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 (1994) 415-437. | MR | Zbl

[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. | MR | Zbl