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PDE's for the Dyson, Airy and Sine processes

Mark Adler (2005)

Annales de l’institut Fourier

In 1962, Dyson showed that the spectrum of a n × n random Hermitian matrix, whose entries (real and imaginary) diffuse according to n 2 independent Ornstein-Uhlenbeck processes, evolves as n non-colliding Brownian particles held together by a drift term. When n → ∞ , the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...

Periodic conservative solutions of the Camassa–Holm equation

Helge Holden, Xavier Raynaud (2008)

Annales de l’institut Fourier

We show that the periodic Camassa–Holm equation u t - u x x t + 3 u u x - 2 u x u x x - u u x x x = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u | t = 0 in H per 1 . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with μ ac = ( u 2 + u x 2 ) d x . The total energy is preserved by the solution.

Proof of the Treves theorem on the KdV hierarchy

Leonid Dickey (2005)

Annales de l’institut Fourier

A new, shorter, proof of the Treves theorem on an algebraic criterion for the first integrals of the KdV hierarchy is given, along with an addition to the theorem.

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