Propagation of chaos for the Burgers equation (Q1082720)

The authors consider the system of diffusions with mean field interaction \[ dX^ n_ i=\frac{1}{n-1}\sum_{j\neq i}e(X^ n_ i,X^ n_ j)dB_ i+\frac{1}{n-1}\sum_{j\neq i}f(X^ n_ i,X^ n_ j)dt,\quad i=1,...,n, \] formulated by \textit{H. P. McKean} [Lect. Differ. Equations 2 (USA 1966-1967), 177-194 (1969; Zbl 0181.444)], in the case \(e=1\) and \(f(x,y)=(\lambda /2)\delta (x-y)\). They prove that for certain chaotic initial distributions the sequence of processes \(\{X^ n\}_ n\) is Z- chaotic, where Z is a diffusion process satisfying \(dZ(t)=dB(t)+(\lambda /2)p_ t(Z(t))dt\) with an initial distribution related to the initial distributions of the \(X^ n_ i\), where \(p_ t(x)\) is the solution of the Burgers equation \(\partial p/\partial t=(1/2)\nabla^ 2p-(\lambda /2)\nabla p^ 2\).