Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

Huang, Jingyu; Lê, Khoa N.; Nualart, David

doi:10.1007/s40072-017-0099-0

Cited by 26 publications

(23 citation statements)

References 14 publications

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“…There is also a literature on well-posedness and regularity theory for (1.1) when f is a distribution that is not necessarily in M + (R d ), though such results tend to be applicable in a more specialized setting as compared with the theory of Dalang [19]; see for example [13,11,30,31,32]. Henceforth, we consider the case that f is a nonnegative-definite, but possibly signed, function of the form, f = h * h, (1.8) where h : R d → R has enough regularity to ensure among other things that the convolution in (1.8) is well defined, and h(x) := h(−x) defines the reflection of h. In this case, (1.2) is equivalent to the elegant formula…”

Section: Introductionmentioning

confidence: 99%

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Chen

Khoshnevisan

Nualart

et al. 2021

Electron. J. Probab.

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Consider a parabolic stochastic PDE of the form ∂tu = 1 2 ∆u + σ(u)η, where u = u(t , x) for t ≥ 0 and x ∈ R d , σ : R → R is Lipschitz continuous and non random, and η is a centered Gaussian noise that is white in time and colored in space, with a possiblysigned homogeneous spatial correlation f . If, in addition, u(0) ≡ 1, then we prove that, under a mild decay condition on f , the process x → u(t , x) is stationary and ergodic at all times t > 0. It has been argued that, when coupled with moment estimates, spatial ergodicity of u teaches us about the intermittent nature of the solution to such SPDEs [1,37]. Our results provide rigorous justification of such discussions.Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field u(t) is mixing for every t > 0; and by (b) giving a quick proof of a conjecture of Conus et al [15] about the "size" of the intermittency islands of u.The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama [42] (see also Dym and McKean [23]) in the simple setting where the nonlinear term σ is a constant function.

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Section: Introductionmentioning

confidence: 99%

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Chen

Khoshnevisan

Nualart

et al. 2021

Electron. J. Probab.

Self Cite

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“…An important ingredient to prove Proposition 1.2 is the following Feynman-Kac formula for the moments of the solution. In the case (H1), this formula is essentially a reformulation of Corollary 4.4 in Huang et al (2017a) and the result for (H2) is new, so we provide in Section 6 a unified proof for both cases.…”

Section: Introductionmentioning

confidence: 93%

“…. > s σ(n) > 0; we refer readers to Hu et al (2015); Huang et al (2017a); Song et al (2020) for the rigorous derivation of (2.3).…”

Section: Mild Solution Now We Definementioning

confidence: 99%

Spatial averages for the parabolic Anderson model driven by rough noise

Nualart¹,

Song²,

Zheng³

2021

ALEA

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In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters H 0 in time and H 1 in space, satisfying H 0 ∈ (1/2, 1), H 1 ∈ (0, 1/2) and H 0 + H 1 > 3/4. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

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“…Since the obstacles are drawn according to a renewal process, our work provides an exemple of localization in a correlated disordered environment. Influence of spatial correlations has been investigated in other contexts such as localization of directed polymers with space-time noise [28], Brownian motion in correlated Poisson potential [29,30,36] as well as Anderson localization, both by mathematicians [25,26,31] and physicists [2,13,43]. In our model however, the extreme value statistics are more relevant than the correlation structure itself.…”

Section: Introductionmentioning

confidence: 99%

Localization of a one-dimensional simple random walk among power-law renewal obstacles

Poisat¹,

Simenhaus²

2022

Preprint

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We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces.

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Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

Cited by 26 publications

References 14 publications

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Spatial averages for the parabolic Anderson model driven by rough noise

Localization of a one-dimensional simple random walk among power-law renewal obstacles

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