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Publication# Stochastic partial differential equations driven by Lévy white noises

Abstract

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

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Related concepts (24)

Space

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

Heat equation

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.

White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal.

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