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

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

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An Extension Problem Related to the Fractional Laplacian

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

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

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

(1982). The local regularity of solutions of degenerate elliptic equations. Communications in Partial Differential Equations: Vol. 7, No. 1, pp. 77-116.

The local regularity of solutions of degenerate elliptic equations

We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

The Inhomogeneous Dirichlet Problem in Lipschitz Domains

u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on ‖u0‖Hs . Equation in (1.1) was derived by Korteweg and de Vries [21] as a model for long wave propagating in a channel. A large amount of work has been devoted to the existence problem for the IVP (1.1). For instance, (see [9], [10]), the inverse scattering method applies to this problem, and, under appropriate decay assumptions on the data, several existence results have been established, see [5],[6],[14],[28],[33]. Another approach, inherited from hyperbolic problems, relies on the energy estimates, and, in particular shows that (1.1) is locally well posed in H(R) for s > 3/2, (see [2],[3],[12],[29],[30],[31]). Using these results and conservation laws, global (in time) well posedness in H(R), s ≥ 2 was established, (see [3],[12],[30]). Also, global in time weak solutions in the energy space H(R) were constructed in [34]. In [13] and [22] a “local smoothing” effect for solutions of (1.1) was discovered. This, combined with the conservation laws, was used in [13] and [22] to construct global in time weak solutions with data in H(R), and even in L(R). In [16], we introduced oscillatory integral techniques, to establish local well posedness of (1.1) in H(R), s > 3/4, and hence, global (in time) well posedness in H(R), s ≥ 1. (In [16] we showed how to obtain the above mentioned result by Picard iteration in an appropriate function space.) In [4] J. Bourgain introduced new function spaces, adapted to the linear operator ∂t+∂ 3 x, for which there are good “bilinear” estimates for the nonlinear term ∂x(u /2). Using these spaces, Bourgain was able to establish local well posedness of (1.1) in H(R) = L(R), and hence, by a conservation

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Carlos E. Kenig

Papers

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

The local regularity of solutions of degenerate elliptic equations

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

The Inhomogeneous Dirichlet Problem in Lipschitz Domains

A bilinear estimate with applications to the KdV equation