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Register a new userEigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion
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In this article, we study high-dimensional behavior of empirical spectral distributions ${L_N(t), tin[0,T]}$ for a class of $Ntimes N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H in(1/2,1)$. For Wigner-type matrices, we obtain almost sure relative compactness of ${L_N(t), tin[0,T]}_{Ninmathbb N}$ in $C([0,T], mathbf P(mathbb R))$ following the approach in cite{Anderson2010}; for Wishart-type matrices, we obtain tightness of ${L_N(t), tin[0,T]}_{Ninmathbb N}$ on $C([0,T], mathbf P(mathbb R))$ by tightness criterions provided in Appendix ref{subset:tightness argument}. The limit of ${L_N(t), tin[0,T]}$ as $Nto infty$ is also characterised.
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We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $Hin (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(cdot,mu) = fastmu(cdot) + g(cdot),quad f,gin B^alpha_{infty,infty}, quad alpha>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.
Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.
We consider eigenvalues of generalized Wishart processes as well as particle systems, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, central limit theorems for the Dysons Brownian motion and the eigenvalues of the Wishart process are recovered under slightly more general initial conditions, and a central limit theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process is obtained.
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform hypoellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution. Our methodology relies heavily on the rough paths structure of the equation.
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform ellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution.
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