Related papers: Free energies and fluctuations for the Unitary Bro…
We study the large N asymptotics of the Brownian motions on the orthogonal, unitary and symplectic groups, extend the convergence in non-commutative distribution originally obtained by Biane for the unitary Brownian motion to the orthogonal…
We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green's functions associated with the process.…
We consider a transient Brownian motion reflected obliquely in a two-dimensional wedge. A precise asymptotic expansion of Green's functions is found in all directions. To this end, we first determine a kernel functional equation connecting…
We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the…
Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent…
We derive new limit theorems for Brownian motion, which can be seen as non-exponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits…
We investigate the unique stationary measure of a positive recurrent reflecting Brownian motion in the upper half-plane, where the direction of reflection is constant on each half-axis. The Laplace transform of the stationary distribution…
It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally,…
After some normalization, the logarithms of the ordered singular values of Brownian motions on $GL(N,\mathbb F)$ with $\mathbb F=\mathbb R, \mathbb C$ form Weyl-group invariant Heckman-Opdam processes on $\mathbb R^N$ of type $A_{N-1}$. We…
We consider exponential functionals of a multi-dimensional Brownian motion with drift, defined via a collection of linear functionals. We give a characterization of the Laplace transform of their joint law as the unique bounded solution, up…
Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \in (0,1)$ split at a…
The first part of this paper is devoted to the Brown measure of the product of the free unitary Brownian motion by an arbitrary free non negative operator. Our approach follows the one recently initiated by Driver-Hall-Kemp though there are…
We present an interesting connection between Brownian motion and magnetism. We use this to determine the distribution of areas enclosed by the path of a particle diffusing on a sphere. In addition, we find a bound on the free energy of an…
Non-Markovian effects upon the Brownian movement of a free particle in the presence as well as in the absence of inertial force are investigated within the framework of Fokker-Planck equations (Rayleigh and Smoluchowski equations). More…
We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the…
We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.
We prove that the convergence of the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion toward Gaussian free fields on the cylinder, as the matrix dimension goes to infinity, holds in…
We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate…
We study the set of points $\mathcal{D}_{n,m}$ around which two independent Brownian motions wind at least $n$ (resp. $m$) times. We prove that its area is asymptotically equivalent, in $L^p$ and almost surely, to…
We study the Yang--Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is…