Abstract: 

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 $\left\{L_{d}(s,t), (s,t)\in[0,S]\times [0,T]\right\}_{d\in\bN}$ of empirical spectral measures of the rescaled matrices is tight on $C([0,S]\times [0,T], \mathcal P(\bR))$ and hence is convergent as $d$ goes to infinity by Wigner's semicircle law.  We also obtain PDEs which are satisfied by the high-dimensional limiting measure.  This is a joint work with Yimin Xiao and Wangjun Yuan. 

个人简介: 宋健，山东大学教授，主要研究领域为随机偏微分方程、随机矩阵、以及统计物理等。