I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

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Lévy Processes and Stochastic Calculus

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

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

Stochastic differential equations and diffusion processes

We consider a one-dimensional stochastic differential equation driven by a compensated Poisson measure. We assume that this equation admits a unique solution We prove that under a strong non-degeneracy condition, for each t ≥ 0, the law of is bounded below by a measure that admits a strictly positive continuous density with respect to the Lebesgue measure on . To this aim, we develop Bismut's approach of the Malliavin calculus for Poisson functionals

http://www.proba.jussieu.fr/pageperso/fournier/a01.pdf

Strict positivity of the density for a poisson driven S.D.E

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka.

Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the problem of the simulation of Levy-driven stochastic differential equations. It is generally impossible to simulate the increments of a Levy-process. Thus in addition to an Euler scheme, we have to simulate approximately these increments. We use a method in which the large jumps are simulated exactly, while the small jumps are approximated by Gaussian variables. Using some recent results of Rio about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady, we derive an estimate for the strong error of this numerical scheme. This error remains reasonnable when the Levy measure is very singular near 0, which is not the case when neglecting the small jumps. 
In the same spirit, we study the problem of the approximation of a Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large jumps.

Simulation and approximation of Levy-driven stochastic differential equations

Smoluchowski's equation: rate of convergence of the Marcus-Lushnikov process

and deterministic initial condition X0(x) 2 C([0, 1]). The symbols _ Nt(dz) and _ W t,x stand respectively for the heuristical Radon±Nikodym density of N (dt, dz) and W (dx, dt) with respect to the Lebesgue measures dt and dt dx. We could also write, with an abuse of notation, _ Nt(dz) dt dx  dxN (dt, dz) and _ Wx, t dt dx  W (dx, dt). We denote by D([0, T ], C([0, 1])) the set of cadlag functions from [0, T ] into C([0, 1]), endowed with the corresponding Skorokhod topology. In this paper, we characterize the Bernoulli 7(1), 2001, 165±190

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Nicolas Fournier

Papers

Strict positivity of the density for a poisson driven S.D.E

Well-posedness of the spatially homogeneous Landau equation for soft potentials

Simulation and approximation of Levy-driven stochastic differential equations

Smoluchowski's equation: rate of convergence of the Marcus-Lushnikov process

Support theorem for the solution of a white-noise-driven parabolic stochastic partial differential equation with temporal Poissonian jumps