We study the spectrum of kinetic Brownian motion in space d×d Hermitian matrices, d≥2. show that eigenvalues stay distinct for all times, and process Λ is a diffusion (i.e. pair (Λ,Λ˙) its derivative Markovian) if only d=2. In large scale time limit, we converges to usual (Markovian) Dyson under suitable normalisation, regardless dimension.

1 Notation In these notes we shall in general use standard notation. For every n ∈ N Bn denotes the Borel algebra on R and if (Ω,F , P ) is a probability space, X : Ω→ R a random variable, then we let X(P ) denote the distribution measure (the image measure) on R of X, e.g. X(P )(A) = P (X−1(A)) for allA ∈ Bn. (1.1) If n ∈ N, we let 〈·, ·〉 denote the canonical inner product on R. Hence for all ...

This is a set of lecture notes based on a graduate course given at the Taught Course Centre in Mathematics in 2011. The course is based on a selection of material from my book with Yuval Peres, entitled Brownian motion, which was published by Cambridge University Press in 2010. 1 Lévy's construction of Brownian motion and modulus of continuity Much of probability theory is devoted to describing...

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [5] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

In finance one of the primary issues is managing risk. Related to this issue and maybe for hedging, investors are naturally interested in the expected values of supremum, infimum, maximum gain and maximum loss of risky assets and the relations between them. Price of a risky asset, stock, can be modeled using Brownian motion and fractional Brownian motion. In this study, we first present the mar...