On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus, Daniel; Khoshnevisan, Davar

doi:10.1007/s00440-010-0333-4

Cited by 53 publications

(90 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, we are left to consider exponents in the region p ∈ (1, 1 + 2/d). In dimension 1, we can use Itô's isometry to calculate second moments, and there are essentially no differences to the estimates (or exact formulae) obtained in the Gaussian case ( [9,12,17]). However, for d ≥ 2, we cannot use Itô's isometry because p is strictly between 1 and 2.…”

Section: The Martingale Casementioning

confidence: 99%

See 1 more Smart Citation

Intermittency for the stochastic heat equation with Lévy noise

Chong¹,

Kevei²

2019

Ann. Probab.

View full text Add to dashboard Cite

We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a (d + 1)-dimensional Lévy space-time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order 1 + 2/d or higher. Intermittency of order p, that is, the exponential growth of the pth moment as time tends to infinity, is established in dimension d = 1 for all values p ∈ (1, 3), and in higher dimensions for some p ∈ (1, 1 + 2/d). The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when p → 1 + 2/d further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail. Summary of resultsFor the stochastic heat equation (1.1), however, there is an important difference that necessitates the development of new techniques for the intermittency analysis. Namely, as soon as Λ contains a non-Gaussian part, the solution to (1.1) will typically have finite moments only up to the order (1 + 2/d) − ǫ, even if Λ itself has moments of all orders or has bounded jump sizes like in the case of a standard Poisson noise, see Theorem 3.1. In particular, as soon as we are in dimension d ≥ 2, the solution has no finite second moment. This is in sharp contrast to the Gaussian case where it is well known that the solution to the stochastic heat equation, if it exists, has finite moments of all orders. And because, as a consequence of the comparison principle in Theorem 3.3, we cannot expect in general that the solution is weakly intermittent of order 1, we are forced to consider moments of non-integer orders in the range (1, 1 + 2/d) ⊆ (1, 2). Therefore, well-established techniques for estimating integer moments of the solution (see [4,9]) do not apply in this setting.This problem can be remedied by an appropriate use of the Burkholder-Davis-Gundy (BDG) inequalities for verifying the intermittency upper bounds, see Theorem 2.4. However, for the corresponding lower bounds, the moment estimates that are available in the literature (including again the BDG inequalities, but also "predictable" versions thereof, see e.g. [21]) do not combine well with the recursive Volterra structure of (1.5). So although these estimates are sharp, we cannot apply them to produce the desired intermittency lower bounds. In order to circumvent this, we use decoupling techniques to establish an -up to our knowledge -new moment lower bound for Poisson stochastic integrals in Lemma 3.4, which we think is of independent interest. With this inequality we then prove the weak intermittency of (1.1) under quite general assumptions. More precisely, if Λ has mean zero, we show in Theorem 3.5 and Theorem 3.6 that we have pth order intermittency for all p ∈ (1, 3) in dimension 1, and for some p ∈ (1, 1 + 2/d) in dimensions d ≥ 2. In the latter case, a small diffu...

show abstract

Section: The Martingale Casementioning

confidence: 99%

“…In particular, we are interested in conditions under which the solution Y to (1.5) exhibits the phenomenon of intermittency. The following definition follows [7], Definition III.1.1, [12], Equations (1.6) and (1.7), and [20], Definition 7.5. Definition 1.1 Let Y be the mild solution to (1.5) and p ∈ (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

Intermittency for the stochastic heat equation with Lévy noise

Chong¹,

Kevei²

2019

Ann. Probab.

View full text Add to dashboard Cite

show abstract

“…Let u ∈ L 2 (Ω; H) and (u n ) n≥1 ⊂ Dom δ such that E u n − u 2 H → 0. Suppose that there exists a random variable G ∈ L 2 (Ω) such that E(δ(u n )F ) → E(GF ) for all F ∈ S, where S is the class of smooth random variables of form (7). Then u ∈ Dom δ and δ(u) = G.…”

Section: Intermittencymentioning

confidence: 99%

“…Recently, there has been a lot of interest in studying the intermittency property of solutions to stochastic partial differential equations, such as the stochastic heat and wave equations. For the former, we refer the reader to [5,7,13,6,17]. On the other hand, intermittency for the solution of the stochastic wave equation driven by a noise which is white in time and has a smoother space correlation than the one considered here was studied in [10,8].…”

Section: Introductionmentioning

confidence: 99%

Intermittency for the Hyperbolic Anderson Model with rough noise in space

Balan

María

Quer-Sardanyons

2017

Stochastic Processes and their Applications

View full text Add to dashboard Cite

show abstract

“…Finally, in Section 6 we study the speed of propagation of intermittent peaks. The propagation of the farthest high peaks was first considered by Conus and Khoshnevisan in [7] for a one-dimensional heat equation driven by a space-time white noise with compactly supported initial condition, where it is shown that there are intermittency fronts that move linearly with time as λt. More precisely, they defined the lower and upper exponential growth indices as follows:…”

Section: Introductionmentioning

confidence: 99%

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

Huang

Lê

Nualart

2017

Stoch PDE: Anal Comp

View full text Add to dashboard Cite

We consider the linear stochastic heat equation on R ℓ , driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter H ∈ ( 1 4 , 1 2 ] in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.2010 Mathematics Subject Classification. 60G15; 60H07; 60H15; 60F10; 65C30.

show abstract

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Cited by 53 publications

References 29 publications

Intermittency for the stochastic heat equation with Lévy noise

Intermittency for the stochastic heat equation with Lévy noise

Intermittency for the Hyperbolic Anderson Model with rough noise in space

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

Contact Info

Product

Resources

About