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Register a new userThe Brown measure of the free multiplicative Brownian motion
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The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $mathsf{GL}(N;mathbb{C}),$ in the sense of $ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $bar{Sigma}_{t},$ which is strictly positive and real analytic on $Sigma_{t}$. This density has a simple form in polar coordinates: [ W_{t}(r,theta)=frac{1}{r^{2}}w_{t}(theta), ] where $w_{t}$ is an analytic function determined by the geometry of the region $Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a shadow of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigners semicircle law and Ginibres circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.
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We consider a family of free multiplicative Brownian motions $b_{s,tau}$ parametrized by a positive real number $s$ and a nonzero complex number $tau$ satisfying $leftvert tau-srightvert leq s,$ with an arbitrary unitary initial condition. We compute the Brown measure $mu_{s,tau}$ of $b_{s,tau}$ and find that it has a simple structure, with a density in logarithmic coordinates that is constant in the $tau$-direction. We also find that all the Brown measures with $s$ fixed and $tau$ varying are related by pushforward under a natural family of maps. Our results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $tau=s.$ We use a version of the PDE method introduced by Driver-Hall-Kemp, but with some significant technical differences.
We compute the Brown measure of $x_{0}+isigma_{t}$, where $sigma_{t}$ is a free semicircular Brownian motion and $x_{0}$ is a freely independent self-adjoint element. The Brown measure is supported in the closure of a certain bounded region $Omega_{t}$ in the plane. In $Omega_{t},$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $x_{0}+isigma_{t}$ by a certain map $Q_{t}:Omega_{t}rightarrowmathbb{R}$ gives the distribution of $x_{0}+sigma_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+isigma_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.
Let ${U^N_t}_{tge 0}$ be a standard Brownian motion on $mathbb{U}(N)$. For fixed $Ninmathbb{N}$ 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 ensemble-averaged spectral measure and to the large-$N$ limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.
We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Itos type (Itos integral, Itos formula, Itos equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem.
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