Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Real and Complex Analysis

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

We study the existence of positive solutions for the nonlinear Schrodinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

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Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)

https://lmb.univ-fcomte.fr/IMG/pdf/albrimo11.pdf

Diffusion as a singular homogenization of the Frenkel-Kontorova model

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

Non-uniqueness of weak solutions for the fractal Burgers equation

We consider the propagation of a flame front in a solid medium with a periodic structure. The model is governed by a free boundary system for the pair" temperature-front. "The front's normal velocity depends on the temperature via a (degenerate) Arrhenius kinetic. It also depends on the front's mean curvature. We show the existence of travelling wave solutions for the full system and consider their homogenization as the period tends to zero. We analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the limiting ratio "curvature coefficient/period." This analysis provides valuable information on the heterogeneous propagation as well.

On the propagation of a periodic flame front by an arrhenius kinetic

Renormalized solutions of the fractional Laplace equation Solutions renormalisées de l'équation de Laplace fractionnaire

We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front's velocity depends on the temperature via a kinetic rate which may degenerate. We show the existence of travelling wave solutions which are bounded and global. Previous results by the same authors (cf. Alibaud, Namah, On the propagation of a periodic flame front by an Arrhenius kinetic,{\it Interfaces Free Bound.,} 2017) were obtained for essentially positively lower bounded kinetics or eventually which have some very weak degeneracy. Here we consider general degenerate kinetics, including in particular those of Arrhenius type which are commonly used in physics.

Nathaël Alibaud

Papers

Diffusion as a singular homogenization of the Frenkel-Kontorova model

Non-uniqueness of weak solutions for the fractal Burgers equation

On the propagation of a periodic flame front by an arrhenius kinetic

Renormalized solutions of the fractional Laplace equation Solutions renormalisées de l'équation de Laplace fractionnaire

Bounded) Traveling Combustion Fronts With Degenerate Kinetics