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

What is the fractional Laplacian? A comparative review with new results

The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a joint theoretical and computational approach to examining their different characteristics by studying solutions of related fractional Poisson equations formulated on bounded domains. 
In this work, we provide new numerical methods as well as a self-contained discussion of state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

What Is the Fractional Laplacian

This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.

On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives

This paper is concerned with a nonlinear integral equation

 $$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$
 where N ≥  1, $${\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}$$
 for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as $${t \to \infty}$$
 .

Asymptotics for a nonlinear integral equation with a generalized heat kernel

We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation

Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space

The initial-value problem for the drift–diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian ( − Δ ) θ / 2 . Large-time behavior of solutions to the drift–diffusion equation with 0 θ ≤ 1 is discussed. When θ > 1 , large-time behavior of solutions is known. However, when 0 θ ≤ 1 , the perturbation methods used in the preceding works would not work. In this paper, the asymptotic expansion of solutions with high-order is derived.

Asymptotic expansion of solutions to the drift–diffusion equation with fractional dissipation

In the author's paper [SIAM J. Math. Anal., 44 (2012), pp. 3786--3805], the author employs the decay estimates of the fundamental solution of the linear dissipative equation to derive the asymptotic expansion for the dissipative equation with the potential term. However, these decay estimates contain an error. The corrected estimates published here prove the main theorems in the original paper.

Erratum: Asymptotic Expansion of Solutions to the Dissipative Equation with Fractional Laplacian

/pdf/existence-and-analyticity-of-solutions-to-the-drift-4y7mi685js.pdf

Masakazu Yamamoto

Papers

Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation

Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space

Asymptotic expansion of solutions to the drift–diffusion equation with fractional dissipation

Erratum: Asymptotic Expansion of Solutions to the Dissipative Equation with Fractional Laplacian

Existence and analyticity of solutions to the drift-diffusion equation with critical dissipation