Parabolic Anderson problem and intermittency

Carmona, René; Molchanov, Stanislav

doi:10.1090/memo/0518

Cited by 247 publications

(314 citation statements)

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“…A discussion of some mathematical problems related to (0.1) can be found in the recent memoir by Carmona and Molchanov (1994). In the present paper we shall be concerned with one particular aspect of (0.1), namely the occurrence of intermittency.…”

Section: Intermittencymentioning

confidence: 99%

“…See Section 0.7 for a description of numerical work. 4 Note that Theorem 2I(3)(iii) implies v (x) = e x p ;(1 + o(1))jxj log jxj] (jxj ! 1 ):…”

Section: Intermittencymentioning

confidence: 99%

“…branching. Other applications are: Fisher-Eigen equation in Darwinian evolution (Ebeling et al (1984)) Burgers' equation with a random force in hydrodynamics (Carmona and Molchanov (1994)). …”

mentioning

confidence: 99%

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Correlation structure of intermittency in the parabolic Anderson model

Gärtner

Hollander

1999

Probab Theory Relat Fields

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Consider the Cauchy problem @u(x t)=@t = Hu(x t) ( x 2 Z Z d t 0) with initial condition u(x 0) 1 and with H the Anderson Hamiltonian H = + . Here is the discrete Laplacian, 2 (0 1) is a di usion constant, and = f (x): x 2 Z Z d g is an i.i.d. random eld taking values in IR. G artner and Molchanov (1990) have shown that if the law o f (0) is nondegenerate, then the solution u is asymptotically intermittent. This means that lim t!1 hu 2 (0 t )i=hu(0 t )i 2 = 1, where h i denotes expectation w.r.t. , and similarly for the higher moments. Qualitatively their result says that, as t increases, the random eld fu(x t): x 2 ZZ d g develops sparsely distributed high peaks, which g i v e the dominant c o n tribution to the moments as they become sparser and higher.In the present paper we study the structure of the intermittent peaks for the special case where the law o f (0) is (in the vicinity of) the double exponential Prob( (0) > s ) = exp ;e s= ] ( s 2 IR). Here 2 (0 1) is a parameter that can be thought of as measuring the degree of disorder in the -eld. Our main result is that, for xed x y 2 Z Z d and t ! 1 , the correlation coe cient o f u(x t) and u(y t) c o n verges to kw k ;2 2 (Z Z) with minimal l 2 -norm). Qualitatively our result says that the high peaks of u have a shape that is a multiple of w relative t o t h e c e n ter of the peak. It will turn out that if the right tail of the law o f (0) is thicker (or thinner) than the double exponential, then the correlation coe cient o f u(x t) and u(y t) c o n verges to x y (resp. the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.1991 Mathematics Subject Classi cation: 60H25, 82C44 (primary), 60F10, 60J15, 60J55 (secondary).

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Section: Intermittencymentioning

confidence: 99%

“…See Section 0.7 for a description of numerical work. 4 Note that Theorem 2I(3)(iii) implies v (x) = e x p ;(1 + o(1))jxj log jxj] (jxj ! 1 ):…”

Section: Intermittencymentioning

confidence: 99%

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Correlation structure of intermittency in the parabolic Anderson model

Gärtner

Hollander

1999

Probab Theory Relat Fields

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“…[3]. It is known that the (random in {W x : x ∈ Z d }) solution satisfies lim t→∞ log u(x, t) t = λ(κ) > 0, P a.s., for nonrandom λ(κ), see [2,4,9,15] and [17] for more details.…”

Section: U(x T) = E X E T 0 Dw X (T−s) (S)mentioning

confidence: 99%

“…It presents a physically relevant model for real world phenomena such as transport of electrons in crystals with impurities and the temperature on the surface of the sun to name just a couple of examples, see [3] and [14]. On the other hand, its analysis has provided mathematical challenges.…”

Section: Introductionmentioning

confidence: 99%

On large deviations for the parabolic Anderson model

Cranston

Gauthier

Mountford

2009

Probab. Theory Relat. Fields

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The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a "law of large numbers", with lim t→∞ 1 t log u(x, t) = λ(κ) and lim n→∞ A n n = α. We then think of αn and λ(κ)t as being the mean of the respective quantities A n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.

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Weighted Stochastic Sobolev Spaces and Bilinear SPDEs Driven by Space–Time White Noise

Nualart

Rozovskiĭ

1997

Journal of Functional Analysis

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In this paper we develop basic elements of Malliavin calculus on a weighted L 2 (0). This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDEs. As an example, we consider a stochastic advection-diffusion equation driven by space-time white noise in R d . It is known that for d>1, this equation has no solutions in L 2 (0). In contrast, it is shown in the paper that in an appropriately weighted L 2 (0) there is a unique solution to the stochastic advection-diffusion equation for any d 1. In addition we present explicit formulas for the Hermite Fourier coefficients in the Wiener chaos expansion of the solution.1997 Academic Press

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Parabolic Anderson problem and intermittency

Cited by 247 publications

References 0 publications

Correlation structure of intermittency in the parabolic Anderson model

Correlation structure of intermittency in the parabolic Anderson model

On large deviations for the parabolic Anderson model

Weighted Stochastic Sobolev Spaces and Bilinear SPDEs Driven by Space–Time White Noise

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