We identify a necessary and sufficient condition for a Lévy white noise to be a tempered distribution. More precisely, we show that if the Lévy measure associated with this noise has a positive absolute moment, then the Lévy white noise almost surely takes values in the space of tempered distributions. If the Lévy measure does not have a positive absolute moment of any order, then the event on which the Lévy white noise is a tempered distribution has probability zero.

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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We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric α-stable noise.

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