Lévy-driven Volterra Equations in Space and Time

Chong, Carsten

doi:10.1007/s10959-015-0662-4

Cited by 20 publications

(43 citation statements)

References 33 publications

(105 reference statements)

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“…Indeed, because σ(Y 0 (t 0 , x 0 )) = 0, there exists by (1.1), which together contradict (3.1)). Therefore, we have E[|σ(Y (t 1 , x 1 ))|] > 0, and because the solution to (1.1) is L r -continuous on (0, ∞) × R d for all r < 1 + 2/d (see [10,Theorem 4.7(2)]), there exist ǫ, δ > 0 such that Y (s, y))|] > ǫ. (5.31) (1) We only carry out the proof for c = 0; the arguments are similar for c > 0 and we leave the details to the reader.…”

Section: Proof Of Theorem 35mentioning

confidence: 99%

Intermittency for the stochastic heat equation with Lévy noise

Chong¹,

Kevei²

2019

Ann. Probab.

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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...

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Section: Proof Of Theorem 35mentioning

confidence: 99%

Intermittency for the stochastic heat equation with Lévy noise

Chong¹,

Kevei²

2019

Ann. Probab.

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“…Next, by [3, (B.5)], we know that G(t; x, y) Cg(t, x − y) for any (t, x, y) ∈ [0, T ] × [0, π] 2 , with g as in (1.4). Consequently, (1) to (4) of Assumption B of [11] are satisfied, and we can apply [11,Theorem 3.5] to obtain the existence of a unique mild solution to (1.1) satisfying (2.3) for all p ∈ (0, 2]. In order to extend this to all p ∈ (2, 3), we notice that the only step in the proof of [11,Theorem 3.5] that uses p 2 is the moment estimate (6.9) given in [11, Lemma 6.1(2)] with respect to the martingale part L M .…”

Section: The Stochastic Heat Equation On An Intervalmentioning

confidence: 99%

“…A predictable random field u = (u(t, x) : (t, x) ∈ [0, T ] × D) is called a mild solution to (1.1) if for all (t, x) ∈ [0, T ] × D, is the solution to the homogeneous version of (1.1). In (1.2) and (1.3), G D denotes the Green's function of the heat operator on D, which for D = R d equals the Gaussian density (1.4) g(t, x) = (4πt) − d 2 e − |x| 2 4t 1 t 0 (when t = 0, we interpret g(0, x) as the Dirac delta function δ 0 (x)), while on a bounded domain D with smooth boundary it has the spectral representation and uniqueness of solutions for equations like (1.1) with Lévy noise have been investigated in [1,2,11,12,31,34]. Already in the linear case with σ(x) ≡ 1, due to the singularity of the Green's kernel on the diagonal x = y near t = 0, each jump of the noise creates a Dirac mass for the solution.…”

Section: Introductionmentioning

confidence: 99%

Path properties of the solution to the stochastic heat equation with Lévy noise

Chong

Dalang

Humeau

2018

Stoch PDE: Anal Comp

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We consider sample path properties of the solution to the stochastic heat equation, in R d or bounded domains of R d , driven by a Lévy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than − d 2 . Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Lévy noises, and to light-as well as heavy-tailed jumps.Let T > 0 and consider, on a stochastic basis (Ω, F, (F t ) t∈[0,T ] , P) satisfying the usual conditions, the stochastic heat equation driven by a Lévy space-time white noise on [0, T ] × D with Dirichlet boundary conditions:where (λ j ) j 1 are the eigenvalues of −∆ with vanishing Dirichlet boundary conditions, and (Φ j ) j 1 are the corresponding eigenfunctions forming a complete orthonormal basis of L 2 (D).In the special case whereL is a Gaussian noise, the existence, uniqueness and regularity of solutions to Equation (1.1) have been extensively studied in the literature, see e.g. [3,10,23,39] for the case of space-time white noise, [15,36,37] for noises that are white in time but colored in space, and [22] for noises that may exhibit temporal covariances as well. In all cases, the mild solution to (1.2) is jointly locally Hölder continuous in space and time, with exponents that depend on the covariance structure of the noise.By contrast, suppose thatL is a Lévy space-time white noise without Gaussian part, that is,where b ∈ R, J is an (F t ) t∈[0,T ] -Poisson random measure on [0, T ] × D × R with intensity dt dx ν(dz), andJ is the compensated version of J. Here ν is a Lévy measure, that is, ν({0}) = 0 and R z 2 ∧ 1 ν(dz) < +∞, and we assume that ν is not identically zero. The existence

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“…While the papers by Albeverio et al (1998) and Applebaum and Wu (2000) remain in the L 2 -framework of Walsh (1986), Saint Loubert Bié (1998 is one of the first to treat Lévydriven stochastic PDEs in L p -spaces with p < 2. The results are extended in Chong (2016) to Volterra equations with Lévy noise, on finite as well as on infinite time domains. In both Saint Loubert Bié (1998) and Chong (2016), one crucial assumption the Lévy noise has to meet is that its Lévy measure, say λ, must satisfy R |z| p λ(dz) < ∞ (1.4) for some p < 1 + 2/d.…”

Section: Introductionmentioning

confidence: 99%

Stochastic PDEs with heavy-tailed noise

Chong

2017

Stochastic Processes and their Applications

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We analyze the nonlinear stochastic heat equation driven by heavy-tailed noise in free space and arbitrary dimension. The existence of a solution is proved even if the noise only has moments up to an order strictly smaller than its Blumenthal-Getoor index. In particular, this includes all stable noises with index $\alpha<1+2/d$. Although we cannot show uniqueness, the constructed solution is natural in the sense that it is the limit of the solutions to approximative equations obtained by truncating the big jumps of the noise or by restricting its support to a compact set in space. Under growth conditions on the nonlinear term we can further derive moment estimates of the solution, uniformly in space. Finally, the techniques are shown to apply to Volterra equations with kernels bounded by generalized Gaussian densities. This includes, for instance, a large class of uniformly parabolic stochastic PDEs.Comment: in press, Stochastic Processes and their Applications, 201

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Lévy-driven Volterra Equations in Space and Time

Cited by 20 publications

References 33 publications

Intermittency for the stochastic heat equation with Lévy noise

Intermittency for the stochastic heat equation with Lévy noise

Path properties of the solution to the stochastic heat equation with Lévy noise

Stochastic PDEs with heavy-tailed noise

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