Moment bounds for some fractional stochastic heat equations on the ball

Nualart, Eulàlia

doi:10.1214/18-ecp147

Cited by 9 publications

(10 citation statements)

References 20 publications

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“…In [7], it was shown that for small λ, the second moment decays exponentially fast while for large λ, the second moment grows exponentially fast. This was sharpened by using precise heat kernel estimates in [12] and [9]. A main aim of this note is to show that if one replaces the usual derivative by a fractional time derivative, this phase transition no longer holds and a more complicated picture emerges.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Remarks on a fractional-time stochastic equation

Foondun

2021

Proc. Amer. Math. Soc.

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We consider a class of fractional time stochastic equation defined on a bounded domain and show that the presence of the time derivative induces a significant change in the qualitative behaviour of the solutions. This is in sharp contrast with the phenomenon showcased in [7] and extented in [12] and [9]. We also show that as one tunes off the fractional in the fractional time derivative, the solution behaves more and more like its usual counterpart.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Remarks on a fractional-time stochastic equation

Foondun

2021

Proc. Amer. Math. Soc.

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“…(1.5) see [10] for the relation between (1.4) and (1.5) and also see [11,16,17,23,27] for the moments and weak intermittency of the solution u to (1.1) on bounded intervals with various boundary conditions. We have set things up so that (1.5) is in fact equivalent to the strict monotonicity of both k → γ(k)/k and k → γ(k)/k.…”

Section: Introduction Background and Main Resultsmentioning

confidence: 97%

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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“…As for bounded domains, Foondun and Nualart [24] considered the stochastic heat equation on an interval (0, L) with space-time white noise and either Dirichlet or Neumann boundary condition, and studied the moments and intermittency properties of the solutions. Nualart [33] and Guerngar and Nane [26] extended the results in [24] to fractional stochastic heat equations with colored noise, but only to the case when the domain is the unit ball in R d plus a Dirichlet boundary condition. In all these works [24,33,26], the initial conditions are assumed to be a bounded function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

Candil¹,

Chen²,

Lee³

2023

Preprint

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We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data ν. We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For C 1,α -domains with Dirichlet condition, the initial data ν is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to |ν|. As an application, we show that the solution is fully intermittent for sufficiently high level λ of noise under the Dirichlet condition, and for all λ > 0 under the Neumann condition.

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Moment bounds for some fractional stochastic heat equations on the ball

Cited by 9 publications

References 20 publications

Remarks on a fractional-time stochastic equation

Remarks on a fractional-time stochastic equation

Dissipation in Parabolic SPDEs

Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

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