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We discuss chains of interacting Brownian motions. Their time reversal invariance is broken because of asymmetry in the interaction strength between left and right neighbor. In the limit of a very steep and short range potential one arrives…
In this paper, we are concerned with the large N limit of linear combinations of the entries of a Brownian motion on the group of N by N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one.…
We describe in detail the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian…
We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are…
We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.
We derive integral formulas, involving the Airy function, for moments of the time a two-sided Brownian motion with parabolic drift attains its maximum.
A new approach to the generalised Brownian motion introduced by M. Bozejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial…
We derive general sufficient conditions for the existence of Riemann-Stieltjes integrals $\int_a^b Yd X$. Our results extend the classical conditions of L.C.Young and improve some recent results that deal with integrals involving a…
In this paper, with the help of a result by Matsumoto--Yor (2000), we prove a Girsanov-type formula for a class of anticipative transforms of Brownian motion which possesses exponential functionals as anticipating factors. Our result…
We establish an integration by parts formula for the semi-group in time $T > 0$ of the kinetic Brownian motion in the Euclidean plane together with its speed in the circle. The stochastic differential equation of our kinetic Brownian motion…
Consider an n-fold integrated Brownian motion. We show that a simple change in time and scale transforms it into a stationary Gaussian process. The collection of stationary processes so constructed not only constitutes an interesting family…
Let $\{U^N_t\}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}(N)$. For fixed $N\in\mathbb{N}$ and $t>0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the…
New, gauge-independent, second-order Lagrangian for the motion of classical, charged test particles is used to derive the corresponding Hamiltonian formulation. For this purpose a Hamiltonian description of the theories derived from the…
We present a general method for simulating an action of $t$ copies of a Haar random unitary for arbitrary compact groups. This construction can be viewed as a representation-theoretic generalization of Zhandry's compressed function oracle…
L\'evy's stochastic area for planar Brownian motion is the difference of two iterated integrals of second rank against its component one-dimen\-sional Brownian motions. Such iterated integrals can be multiplied using the sticky shuffle…
Brownian motion is a building block in modern probability theory. In this paper, we describe a formalization of Brownian motion using the Lean theorem prover. We build on the existing measure-theoretic foundations in Lean's mathematical…
The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar…
Let X^{1}, X^{2} be two independent (two-sided) fractional Brownian motions having the same Hurst parameter H in (0,1), and let Y be a standard (one-sided) Brownian motion independent of (X^{1},X^{2}). In dimension 2, fractional Brownian…
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 begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of [25],[10] and others, and then formulate and study in details (with a number of interesting examples) a definition…