On the multifractal local behavior of parabolic stochastic PDEs

Huang, Jingyu; Khoshnevisan, Davar

doi:10.1214/17-ecp86

Cited by 6 publications

(3 citation statements)

References 19 publications

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“…A standard method for finding such points is to appeal to the theory of limsup random fractals [11] and adapt it to the present small-ball setting for SPDEs. For largeball problems, this adaptation was done in [8], and we feel that similar methods will yield exceptional points x ∈ T for which the rate const…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To be sure of the order of the quantifiers, we note that Corollary 1.2 says that for every non-random point x ∈ R there exists a P-null set off of which (1.4) holds. We may view such points x as points of [relatively] "flat growth," for example as compared with points where iterated logarithm fluctuations are observed; see [8]. Corollary 1.2 and Fubini's theorem together show that the collection of all points x ∈ T that satisfy (1.4) has full Lebesgue/Haar measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Dissipation in Parabolic SPDEs

et al. 2020

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The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Dissipation in Parabolic SPDEs

et al. 2020

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“…Local structure of the SHE and KPZ were explored in [KSXZ13,FKM15,HK17]. In [KSXZ13], the authors considered a generalization of the SHE in (1.1):…”

Section: Introductionmentioning

confidence: 99%

Temporal increments of the KPZ equation with general initial data

Das¹

2022

Preprint

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We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation H f (t, x) started with initial data f . In this article, we study the sample path properties of the KPZ temporal process H f t := H f (t, 0). We show that for a very large class of initial data which includes any deterministic continuous initial data that grows slower than parabola (f (x) ≪ 1 + x 2 ) as well as narrow wedge and stationary initial data, temporal increments of H f t are well approximated by increments of a fractional Brownian motion of Hurst parameter 1 4 . As a consequence, we obtain several sample path properties of KPZ temporal properties including variation of the temporal process, law of iterated logarithms, modulus of continuity and Hausdorff dimensions of the set of points with exceptionally large increments.

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Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Wang,

Xiao

2024

Theor. Probability and Math. Statist.

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Consider the stochastic partial differential equation ∂ ∂ t u t ( x ) = − ( − Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where − ( − Δ ) α 2 -(-\Delta )^{\frac {\alpha }{2}} denotes the fractional Laplacian with power α 2 ∈ ( 1 2 , 1 ] \frac {\alpha }{2}\in (\frac 12,1] , and the driving noise F ˙ \dot F is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient u t + ε ( x ) − u t ( x ) u_{t+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x}) at any fixed t > 0 t > 0 and x ∈ R d \boldsymbol {x}\in \mathbb R^d , as ε ↓ 0 {\varepsilon }\downarrow 0 . As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the q q -variations of the temporal process { u t ( x ) } t ≥ 0 \{u_t(\boldsymbol {x})\}_{ t \ge 0} of the solution, where x ∈ R d \boldsymbol {x}\in \mathbb R^d is fixed.

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On the multifractal local behavior of parabolic stochastic PDEs

Cited by 6 publications

References 19 publications

Dissipation in Parabolic SPDEs

Dissipation in Parabolic SPDEs

Temporal increments of the KPZ equation with general initial data

Temporal properties of the stochastic fractional heat equation with spatially-colored noise

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