A propagation of chaos result for a system of particles with moderate interaction
Let (\(\Omega\),\({\mathcal F},{\mathcal F}_ t,P)\) be a filtered probability space, endowed with n independent \({\mathbb{R}}^ d\)-valued Brownian motions \(B^ 1,...,B^ n\), and \({\mathbb{R}}^{nd}\)-valued r.v. \((X^ i_ 0)_{1\leq i\leq n}\), \({\mathcal F}_ 0\)-measurable, whose law \(u^ n\) on \({\mathbb{R}}^{nd}\) is symmetric. Consider the following system of particles: \[ dX_ t^{i,n}=F(X_ t^{i,n},V^ n*\mu_ t(X_ t^{i,n}))dt+dB^ i_ t,\quad X_ 0^{i,n}=X^ i_ 0,\quad 0\leq t\leq T,\quad 1\leq i\leq n, \] where \(\mu^ n=n^{- 1}\sum^{n}_{i=1}\delta_{X^{i,n}}\), \(V^ n(.)=n^{\beta}V^ 1(n^{\beta /d}.)\), \(\beta >0\) for a given bounded density function \(V^ 1.\) The laws \(P^ n\) of \((X^{1,n},...,X^{n,n})\) are called \(P^ 0\)- chaotic iff \(\mu^ n\) converge in law to \(P^ 0\). This is an equivalent formulation of the propagation of chaos due to Sznitman. The main result of the paper is as follows: Under some hypotheses on F and \(V^ 1\), if the initial laws \(u^ n\) of \((X_ 0^{1,n},...,X_ 0^{n,n})\) are \(u^ 0\)-chaotic, then so are the laws of the system \((X^{1,n},...,X^{n,n})\).
scientific article; zbMATH DE number 4030681