On the chaotic character of the stochastic heat equation, before the onset of intermitttency

Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar

doi:10.1214/11-aop717

Cited by 66 publications

(155 citation statements)

References 26 publications

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“…Of course, this proves that 12) with room to spare. Hence, Condition (4.34) is verified since the Y i 's are independent.…”

Section: Peaks Of the Ornstein-uhlenbeck Processmentioning

confidence: 75%

“…It follows easily from this convention that the lim sup law (8.18) is a consequence of (8.19). Conus et al [12] proved that the lim sup in (8.18) is strictly positive and finite a.s., and the evaluation of the lim sup is contained in the recent work of Chen [8]. We have included (8.18) merely to highlight the fact that the tall peaks of u t (x) are of order exp{γ t log x } for a constant γ > 0, and hence that (8.19) is indeed a multifractal description of the tall peaks of u t (x).…”

Section: The Main Resultsmentioning

confidence: 99%

“…Since S n contains at most const · exp(nd) upright boxes of sidelength one, it follows that 12) for all n 1 sufficiently large. In particular,…”

Section: General Boundsmentioning

confidence: 99%

“…Lemma 2.4 is the large-scale/macroscopic analogue of the following wellknown fact: If f : E → R is locally Lipschitz continuous, then 12) where "dim H " denotes the usual [microscopic] Hausdorff dimension in R d . Let us, however, observe that (2.8) does not hold when f is only Lipschitz.…”

Section: Dimension and Densitymentioning

confidence: 99%

“…In this section, we plan to study the set of points x > exp(e) at which the solution u t (x) exceeds certain high peaks. For the parabolic Anderson modelthat is when σ(z) = z for all z ∈ R-Conus et al [12] have demonstrated that, for every t > 0, the tall peaks of x → h t (x) are of rough height (log |x|) 2/3 as |x| → ∞. Specifically, they have proved that ] have also proved that the function (log x) 2/3 fails to correctly gauge the height of the tall peaks of h t (x) for general nonlinearities σ.…”

Section: Peaks Of a Nonlinear Stochastic Heat Equationmentioning

confidence: 99%

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Intermittency and multifractality: A case study via parabolic stochastic PDEs

Khoshnevisan¹,

Kim²,

Xiao³

2017

Ann. Probab.

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Let ξ denote space-time white noise, and consider the following stochastic partial differential equations: (i)u = 1 2 u ′′ + uξ, started identically at one; and (ii)Ż = 1 2 Z ′′ + ξ, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes.We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on R+ × R d with d 2. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question.As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein-Uhlenbeck process on R are multifractal.Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S.J. Taylor [3,4]. We expand on aspects of the Barlow-Taylor theory, as well.

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“…Of course, this proves that 12) with room to spare. Hence, Condition (4.34) is verified since the Y i 's are independent.…”

Section: Peaks Of the Ornstein-uhlenbeck Processmentioning

confidence: 75%

Section: The Main Resultsmentioning

confidence: 99%

“…Since S n contains at most const · exp(nd) upright boxes of sidelength one, it follows that 12) for all n 1 sufficiently large. In particular,…”

Section: General Boundsmentioning

confidence: 99%

Section: Dimension and Densitymentioning

confidence: 99%

Section: Peaks Of a Nonlinear Stochastic Heat Equationmentioning

confidence: 99%

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Intermittency and multifractality: A case study via parabolic stochastic PDEs

Khoshnevisan¹,

Kim²,

Xiao³

2017

Ann. Probab.

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Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

Conus

Joseph

Khoshnevisan

et al. 2013

Springer Proceedings in Mathematics &Amp; Statistics

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We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which had only been carefully studied in some particular cases so far. Then, we establish a comparison principle for the solution, following the ideas of Mueller. We think it is of particular interest to obtain such a result for an hyperbolic equation. Finally, using the results mentioned above, we aim to show that the solution exhibits a chaotic behavior, in a similar way as was established in [9] for the heat equation. We study the two cases where 1. the initial conditions have compact support, where the global maximum of the solution remains bounded and 2. the initial conditions are bounded away from 0, where the global maximum is almost surely infinite. Interesting estimates are also provided on the behavior of the global maximum of the solution.Keywords and phrases. Stochastic partial differential equations, wave equation, intermittency.where σ : R → R is a globally Lipschitz function with Lipschitz constant Lip σ , the noise (Ẇ (t, x), t 0, x ∈ R) is space-time white noise and κ > 0. We consider non-random, bounded and measurable initial condition u 0 : R → R + and initial derivative v 0 : R → R.Equation (1.1) has been studied by Carmona and Nualart [6] and Walsh [23]. There are also results available in the more delicate setting where x ∈ R d for d > 1; see Conus and Dalang [8], Dalang [13], Dalang and Frangos [14], and Dalang and Mueller [15].The parabolic equivalent to equation (1.1) is a well-studied family of Stochastic PDEs. In particular, it contains the well-known Parabolic Anderson Model. For more about this parabolic family, we refer the reader to Foondun and Khoshnevisan [18].

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A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

Khoshnevisan

Kim

Xiao

2018

Commun. Math. Phys.

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It is generally argued that the solution to a stochastic PDE with multiplicative noise-such asu = 1 2 u ′′ + uξ, where ξ denotes space-time white noise-routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989;. The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

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On the chaotic character of the stochastic heat equation, before the onset of intermitttency

Cited by 66 publications

References 26 publications

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

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