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Register a new userStrong Convergence of Unitary Brownian Motion
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The Brownian motion $(U^N_t)_{tge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{tge 0}$ as $Ntoinfty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.
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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 consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $eta_* geq N^{-2/3}$ and no outliers, then after times $t gg sqrt{ eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-ErdH{o}s-Yau-Yin] and the energy estimate of [Bourgade-ErdH{o}s-Yau].
To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. This process generalizes both the well known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractional Brownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long range dependence properties of this process.
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
Nils Tongring (1987) proved sufficient conditions for a compact set to contain $k$-tuple points of a Brownian motion. In this paper, we extend these findings to the fractional Brownian motion. Using the property of strong local nondeterminism, we show that if $B$ is a fractional Brownian motion in $mathbb{R}^d$ with Hurst index $H$ such that $Hd=1$, and $E$ is a fixed, nonempty compact set in $mathbb{R}^d$ with positive capacity with respect to the function $phi(s) = (log_+(1/s))^k$, then $E$ contains $k$-tuple points with positive probability. For the $Hd > 1$ case, the same result holds with the function replaced by $phi(s) = s^{-k(d-1/H)}$.
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