Dissipation and high disorder

Chen, Le; Cranston, M.; Khoshnevisan, Davar; Kim, Kun-Woo

doi:10.1214/15-aop1040

Cited by 6 publications

(3 citation statements)

References 25 publications

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“…One may see a similar result in our earlier paper [16] where we consider a stochastic heat equation driven by space-time white noise on the one-dimensional torus R/Z rather than on R. Because R is not compact, we need to make significant modifications to the method of [16] especially when we estimate moments; see §3.1. Once we are able to prove that sup x∈R v(t , x) can be controlled by v(t) L 1 (R) , we appeal to a known result about dissipation of the total mass of the solution (see Chen, Cranston, Khoshnevisan, and Kim [4]) to prove Theorem 1.2; this is done in §4. Finally, we combine the results from § §2-4 in order to verify Theorem 1.1 in §5.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the valleys of the stochastic heat equation

Khoshnevisan¹,

Kim²,

Mueller³

2022

Preprint

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We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution.Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as t → ∞, and we establish an upper bound of exp{−const • t 1/3 } for the rate of decay. We demonstrate also that the length of a valley grows at least as exp{+const • t 1/3 } as t → ∞.Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the valleys of the stochastic heat equation

Khoshnevisan¹,

Kim²,

Mueller³

2022

Preprint

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“…These distinctive features of the peaks and valleys in PAM are expected to be present in the case of parabolic Anderson equation. In fact, [CCKK17] showed that if the initial data decays at infinity faster than the Gaussian kernel, then the total mass of the solution of the parabolic Anderson equation dissipates, that is, it vanishes sub-exponentially as t → ∞. When (1.1) is considered on the torus instead of R, [KKMS20] proved among other things that the supremum of the solution is localized in space.…”

Section: Introductionmentioning

confidence: 99%

Fractal Geometry of the Valleys of the Parabolic Anderson Equation

Ghosal¹,

Yi²

2021

Preprint

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We study the macroscopic fractal properties of the deep valleys of the solution of the (1 + 1)-dimensional parabolic Anderson equationwhere Ẇ is the time-space white noise and 0 < infx∈R u0(x) ≤ sup x∈R u0(x) < ∞. Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are macroscopically monofractal. In fact, the macroscopic Hausdorff dimension (introduced by Barlow and Taylor [BT89, BT92]) of the valleys undergoes a phase transition at a point which does not depend on the initial data. The key tool of our proof is a lower bound to the lower tail probability of the parabolic Anderson equation. Such lower bound is obtained for the first time in this paper and will be derived by utilizing the connection between the parabolic Anderson equation and the Kardar-Parisi-Zhang equation. Our techniques of proving this lower bound can be extended to other models in the KPZ universality class including the KPZ fixed point.

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“…For Eq. (1.1a) in the whole space, the weak intermittency of the solution both on the real line ( [17]) and on the lattice ( [9,18,20]) has been studied. For the continuous Eq.…”

Section: Introductionmentioning

confidence: 99%

Weak intermittency and second moment bound of a fully discrete scheme for stochastic heat equation

Chen¹,

Dang²,

Hong³

2021

Preprint

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In this paper, we first prove the weak intermittency, and in particular the sharp exponential order Cλ 4 t of the second moment of the exact solution of the stochastic heat equation with multiplicative noise and periodic boundary condition, where λ > 0 denotes the level of the noise. In order to inherit numerically these intrinsic properties of the original equation, we introduce a fully discrete scheme, whose spatial direction is based on the finite difference method and temporal direction is based on the θ-scheme. We prove that the second moment of numerical solutions of both spatially semi-discrete and fully discrete schemes grows at least as exp{Cλ 2 t} and at most as exp{Cλ 4 t} for large t under natural conditions, which implies the weak intermittency of these numerical solutions. Moreover, a renewal approach is applied to show that both of the numerical schemes could preserve the sharp exponential order Cλ 4 t of the second moment of the exact solution for large spatial partition number., ∀p > 0. The random field u is called intermittent (also called fully intermittent) if for all x, the mapping p → γp (x)/p is strictly increasing on p ∈ [1, ∞). This Key words and phrases. Stochastic heat equation • Weak intermittency • Sharp exponential order • Numerical scheme • Discrete Green function.

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Dissipation and high disorder

Cited by 6 publications

References 25 publications

On the valleys of the stochastic heat equation

On the valleys of the stochastic heat equation

Fractal Geometry of the Valleys of the Parabolic Anderson Equation

Weak intermittency and second moment bound of a fully discrete scheme for stochastic heat equation

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