Related papers: On the singular values of complex matrix Brownian …
We find a representation of the integral of a Gauss-Markov process in the interval [0, t], in terms of Brownian motion. Moreover, some connections with first-passagetime problems are discussed, and some examples are reported.
This short note is motivated by a recently discovered connection between a drift-diffusion process in $n$-dimensional Euclidean space with a divergence-free drift sampled from a stationary and isotropic Gaussian ensemble of critical scaling…
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum mechanical…
In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t), t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic…
This paper introduces the Neural-Brownian Motion (NBM), a new class of stochastic processes for modeling dynamics under learned uncertainty. The NBM is defined axiomatically by replacing the classical martingale property with respect to…
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear…
In this paper we investigate the Quantum Brownian motion of a point particle induced by quantum vacuum fluctuations of a massless scalar field in (3 + 1)-dimensional Minkowski spacetime with distinct conditions (Dirichlet, Neumann, mixed…
We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only…
Pitman's theorem states that if {Bt, t $\ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$\le$t Bs, t $\ge$ 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever…
Consider an $n\times n$ Hermitean matrix valued stochastic process $\{H_t\}_{t\geq 0}$ where the matrix elements evolve according to Ornstein-Uhlenbeck processes. It is well known that the eigenvalues perform a so called Dyson Brownian…
We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This…
The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, featuring the anomalous diffusion property characterized by the diffusion exponent. It is a Gaussian, self-similar process with independent…
A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the…
We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and the bottom curve are equal to Fredholm determinants whose kernel we give explicitly. In the…
Matrix Dirichlet processes, in reference to their reversible measure, appear in a natural way in many different models in probability. Applying the language of diffusion operators and the method of boundary equations, we describe Dirichlet…
The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in…
We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and…
We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the…
Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable $x$) as the system, {\it nonlinearly} coupled to an environment of $N$…
We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast…