Propagation of chaos for diffusing particles of two types with singular mean field interaction (Q1083130)

The authors discuss a system \((X_ i,Y_ i)\), \(i=1,2,...,n\) of particles of two types, subject to a pair interaction. Namely, \[ X_ i=X_ i(0)+B_ i(t)+\int^{t}_{0}ds\{(1/n)\sum^{n}_{l=1}f(X_ i(s)-Y_ l(s))+v(X_ i(s))\}+\Phi_ i(t) \] and \[ Y_ j=Y_ j(0)+B_ j'(t)+\int^{t}_{0}ds\{(-1/n)\sum^{n}_{l=1}f(X_ l(s)- Y_ j(s))+v(Y_ j(s))\}-\Psi_ j(t) \] i,j\(=1,2,...,n\), where \(B_ i(t)\), \(B_ j'(t)\) are mutually independent Brownian motions and \(\Phi_ i\), \(\Psi_ j\) are chosen in such a way that \(X_ i\) (resp. \(Y_ j)\) is a reflecting process on [0,\(\infty)\) (resp. (-\(\infty,0])\) repelled by \(Y_ l\) with the interaction 1/n f(X\({}_ i-Y_ l)\) (resp. -1/n f(X\({}_ l-Y_ j))\) under the influence of an environment potential \(v(X_ i)\) (resp. \(v(Y_ j))\). It is shown that \((X_ i,Y_ j)\) converges in law, as \(n\to \infty\), to (X,Y) which is a solution of another system of stochastic differential equations. It is allowed for f(x)\(\uparrow \infty\) as \(x\downarrow 0\).