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.)

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case alpha \in (0,1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.

Asymptotic properties of entropy solutions to fractal Burgers equation

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, \({\triangle^{\alpha/2}}\) for \({\alpha \in (0,2)}\). Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on α in the BV-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits \({\alpha \downarrow 0}\) and \({\alpha \uparrow 2}\). In the limit \({\alpha \uparrow 2}\), \({\triangle^{\alpha/2}}\) converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231–251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).

https://hal.archives-ouvertes.fr/hal-00841159/document

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

https://lmb.univ-fcomte.fr/IMG/pdf/alanbe10.pdf

Renormalized solutions of the fractional Laplace equation

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g., the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover and extend to new equations a large part of the known error estimates in the literature.

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

This paper is devoted to the study of semi-linear parabolic equations whose principal term is fractional, i.e. is integral and eventually singular. A typical example is the fractional Laplace operator. This work sheds light on the fact that, if the initial datum is not bounded, assumptions on the non-linearity are closely related to its behavior at infinity. The sub-linear and super-linear cases are first treated by classical techniques. We next present a third original case: if the associated first order Hamilton-Jacobi equation is such that perturbations propagate at finite speed, then the semi-linear parabolic equation somehow keeps memory of this property. By using such a result, locally bounded initial data that are merely integrable at infinity can be handled. Next, regularity of the solution is proved. Eventually, strong convergence of gradients as the fractional term disappears is proved for strictly convex non-linearity.

/pdf/fractional-semi-linear-parabolic-equations-with-unbounded-1pse1luyc7.pdf

Nathaël Alibaud

Papers

Asymptotic properties of entropy solutions to fractal Burgers equation

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

Renormalized solutions of the fractional Laplace equation

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

Fractional semi-linear parabolic equations with unbounded data