Celebratio Mathematica

M. Yor: Some aspects of Brownian motion, part 2: Some recent martingale problems. Lectures in Mathematics ETH Zürich. Birkhäuser (Basel), 1997. MR 1442263 Zbl 0880.60082 book

H. Matsumoto and M. Yor: “An analogue of Pitman’s \( 2M{-}X \) theorem for exponential Wiener functionals, I: A time-inversion approach,” Nagoya Math. J. 159 (2000), pp. 125–166. MR 1783567 Zbl 0963.60076 article

H. Matsumoto and M. Yor: “A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration,” Osaka J. Math. 38 : 2 (2001), pp. 383–398. MR 1833628 Zbl 0981.60078 article

H. Matsumoto and M. Yor: “An analogue of Pitman’s \( 2M{-}X \) theorem for exponential Wiener functionals, II: The role of the generalized inverse Gaussian laws,” Nagoya Math. J. 162 (June 2001), pp. 65–86. MR 1836133 Zbl 0983.60075 article

N. O’Connell and M. Yor: “Brownian analogues of Burke’s theorem,” Stochastic Process. Appl. 96 : 2 (December 2001), pp. 285–304. MR 1865759 Zbl 1058.60078 article

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke’s theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.

N. O’Connell and M. Yor: “A representation for non-colliding random walks,” Electron. Commun. Probab. 7 (2002), pp. 1–12. Article no. 1. MR 1887169 Zbl 1037.15019 article

We define a sequence of mappings \[ \Gamma_k:D_0(\mathbb{R}_+)^k\to D_0(\mathbb{R}_+)^k \] and prove the following result: Let \( N_1,\dots \), \( N_n \) be the counting functions of independent Poisson processes on \( \mathbb{R}_+ \) with respective intensities \( \mu_1 \lt{} \) \( \mu_2 \lt \cdots \lt \mu_n \). The conditional law of \( N_1,\dots \), \( N_n \), given that \[ N_1(t)\leq \cdots \leq N_n(t)\quad\textrm{for all } t\geq 0, \] is the same as the unconditional law of \( \Gamma_n(N) \). From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov [2001] and Gravner, Tracy and Widom [2001], which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman’s representation for the 3-dimensional Bessel process.

C. Donati-Martin, Y. Doumerc, H. Matsumoto, and M. Yor: “Some properties of the Wishart processes and a matrix extension of the Hartman–Watson laws,” Publ. Res. Inst. Math. Sci. 40 : 4 (2004), pp. 1385–1412. MR 2105711 Zbl 1076.60067 article

P. Bourgade, C. P. Hughes, A. Nikeghbali, and M. Yor: “The characteristic polynomial of a random unitary matrix: A probabilistic approach,” Duke Math. J. 145 : 1 (2008), pp. 45–69. MR 2451289 Zbl 1155.15025 ArXiv 0706.0333 article

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin–Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [2000] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [2000] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results.

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Marc Yor

Marc Yor et les matrices aléatoires

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