On the propagation of chaos for diffusion processes with drift coefficients not of average form
The propagation of chaos result is established for the diffusion \[ dX_ i^{(n)}(t)=dB_ i(t)+b^{(n)}[X_ i^{(n)}(t),U^{(n)}(t)]d\quad t,\quad i\leq n, \] with general form of the drift coefficient when \(n\to \infty\), where \(U^{(n)}\) is the empirical distribution of \((X_ 1^{(n)},...,X_ n^{(n)})\), \((B_ i)_{i\geq 1}\) are independent Wiener processes, under rather weak convergence assumption \(b^{(n)}[\cdot,u_ n]\to b[\cdot,u]\) when \(u_ n\to u.\) The main theorem states: (a) \(U^{(n)}(t)\to^{p}u(t)\) where \(u(t)=Law (X(t))\), and X(t) is the strong solution of the McKean-Vlasov SDE \[ (*)\quad X(t)=X(0)+B(t)+\int^{t}_{0}b[X(s),u(s)]ds; \] (b) for every \(m<\infty\), \((X_ 1^{(n)}(t),...,X_ m^{(n)}(t))\to (X_ 1(t),...,X_ m(t))\) in law, where \((X_ i)\) are independent solutions of (*).
scientific article; zbMATH DE number 4038907