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We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence $left{L_{d}(s,t), (s,t)in[0,S]times [0,T]right}_{dinmathbb N}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]times [0,T], mathcal P(mathbb R))$ and hence is convergent as $d$ goes to infinity by Wigners semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.
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We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{alpha, beta}$ with Hurst parameter $(alpha, beta) in (0,1)^2$. When $0<alpha leq 1-frac{1}{2q}$ or $0<beta leq 1-frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $qgeq 2$, while for $1-frac{1}{2q}<alpha, beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
For real symmetric and complex Hermitian Gaussian processes whose values are $dtimes d$ matrices, we characterize the conditions under which the probability that at least $k$ eigenvalues collide is positive for $2le kle d$, and we obtain the Hausdorff dimension of the set of collision times.
We introduce $n$-parameter $Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each time parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of $n$ linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) connection to fourth order linear PDEs. Our BTBS does not belong to the classical theory of random fields; and to prove our new PDEs connections, we generalize our BTP approach in cite{Abtp1,Abtp2} and we mix it with the Brownian sheet connection to a linear PDE system, which we also give along with its corresponding nonlinear second order PDE and $2n$-th order linear PDE. In addition, we introduce the $n$-parameter $d$-dimensional linear Kuramoto-Sivashinsky (KS) sheet kernel (or transition density); and we link it to an intimately connected system of new linear Kuramoto-Sivashinsky-variant interacting PDEs, generalizing our earlier one parameter imaginary-Brownian-time-Brownian-angle kernel and its connection to the KS PDE. The interactions here mean that our PDEs systems are to be solved for a family of functions, a feature shared with well known fluids dynamics models. The interacting PDEs connections established here open up another new fundamental front in the rapidly growing field of iterated-type processes and their connections to both new and important higher order PDEs and to some equivalent fractional Cauchy problems. We connect the BTBS fourth order interacting PDEs system given here with an interacting fractional PDE system and further study it in another article.
We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber - which can be seen as a space time conditioned Brownian motion - is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.
We have proved in a previous paper that a space-time Brownian motion conditioned to remain in a Weyl chamber associated to an affine Kac-Moody Lie algebra is distributed as the radial part process of a Brownian sheet on the compact real form of the underlying finite dimensional Lie algebra, the radial part being defined considering the coadjoint action of a loop group on the dual of a centrally extended loop algebra. We present here a very brief proof of this result based on a time inversion argument and on elementary stochastic differential calculus.
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