On eigenvalues of the Brownian sheet matrix

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.