Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

We present here a new method for solving minimization problems in unbounded domains. We first derive a general principle showing the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions. The proof is based upon a compactness lemma obtained with the help of the notion of concentration function of a measure. We give various applications to problems arising in Mathematical Physics.

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The concentration-compactness principle in the calculus of variations. The locally compact case, part 1

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrodinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

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Endpoint Strichartz estimates

The concentration-compactness principle in the calculus of variations. The locally compact case

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Preliminaries The linear Schrodinger equation The Cauchy problem in a general domain The local Cauchy problem Regularity and the smoothing effect Global existence and finite-time blowup Asymptotic behavior in the repulsive case Stability of bound states in the attractive case Further results Bibliography.

Semilinear Schrodinger Equations

We present a general method which enables us to prove the orbital stability of some standing waves in nonlinear Schrodinger equations. For example, we treat the cases of nonlinear Schrodinger equations arising in laser beams, of time-dependent Hartree equations ....

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Orbital stability of standing waves for some nonlinear Schrödinger equations

The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s

Contents Introduction 1. Preliminary results 2. m-dissipative operators 3. The Hille-Yosida-Phillips Theorem and applications 4. Nonhomogeneous equations and abstract semilinear problems 5. The heat equation 6. The Klein-Gordon equation 7. The Schrodinger equation 8. Bounds on global solutions 9. The invariance principle and some applications 10. Stability of stationary solutions 11. Comments Bibliography Notation Index

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Thierry Cazenave

Papers

Semilinear Schrodinger Equations

Orbital stability of standing waves for some nonlinear Schrödinger equations

The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s

An Introduction to Semilinear Evolution Equations

A nonlinear heat equation with singular initial data