Unified View on L\'evy White Noises: General Integrability Conditions and Applications to Linear SPDE

There exists several ways of constructing L\'evy white noise, for instance are as a generalized random process in the sense of I.M. Gelfand and N.Y. Vilenkin, or as an independently scattered random measure introduced by B.S. Rajput and J. Rosinski. In this article, we unify those two approaches by extending the L\'evy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for L\'evy white noises, thereby maximally extending their domain of definition. Based on this connection, we provide new criteria for the practical determination of this domain of definition, including specific results for the subfamilies of Gaussian, symmetric-$\alpha$-stable, Laplace, and compound Poisson noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a L\'evy white noise.Comment: 43 page

Path properties of the solution to the stochastic heat equation with L\'evy noise

We consider sample path properties of the solution to the stochastic heat equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a L\'evy 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 c\`adl\`ag modification in fractional Sobolev spaces of index less than $-\frac d 2$. 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 L\'evy noise such that noises with a smaller index entail continuous sample paths, while L\'evy 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 L\'evy noises, and to light- as well as heavy-tailed jumps

Pressure dependent friction on granular slopes close to avalanche

We investigate the sliding of objects on an inclined granular surface close to the avalanche threshold. Our experiments show that the stability is driven by the surface deformations. Heavy objects generate footprint-like deformations which stabilize the objects on the slopes. Light objects do not disturb the sandy surfaces and are also stable. For intermediate weights, the deformations of the surface destabilize the objects and generate sliding. A characteristic pressure for which the solid friction is minimal is evidenced. Applications to the locomotion of devices and animals on sandy slopes as a function of their mass are proposed