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

Abstract

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

Intermittency for the wave equation with Lévy white noise

“…Our second goal is to show that the solution of equation ( 7) has a càdlàg modification, when viewed as a process with values in a suitable fractional Sobolev space. A similar phenomena has been studied in [13] for the heat equation. The starting point of this analysis is a quick look at the behavior of G t when t = 0.…”

“…By Theorem A.4 of [13], there exists a probability measure Q on (Ω, F ) which is equivalent to P, such that dQ dP is bounded,…”

“…(1 + |ξ| 2 ) r dξ → 0 as n → ∞, using the (stochastic) dominated convergence theorem given by Theorem A.1 of [13], applied to the random measure L M (under the probability measure Q). In our case, the integrand σ (n),N 1 K → 0 converges pointwise to 0 as n → ∞, and is bounded by the random function |σ(u N )1 K | which belongs to L 1,2 (L M , Q).…”

The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise

“…As the focus of this paper is on the global irregularity of the solution, we will assume (2.2) in all what follows. In particular, by [5,Theorem 3.5], if there exists 0…”

Normal approximation of the solution to the stochastic heat equation with Lévy noise

“…for all t ≤ T , then by Theorem 2.5 in [7], u ε has a càdlàg modification in H −r ([0, π]) for any r > 1/2, which will be denoted by u ε = (u ε t ) t≤T throughout this work. Similarly, by the identification (2.10) and Corollary 3.4 in [27], the mild solution u to (2.8) has a continuous modification u in H −r ([0, π]) for each r > 1/2.…”