We study diffusions with bounded pairwise interaction. We show for the
first time propagation of chaos on arbitrary time horizons in a stronger
-based
distance, as opposed to the usual Wasserstein or relative entropy
distances. The estimate is based on iterating inequalities derived from the
BBGKY hierarchy and does not follow directly from bounds on the full
-particle
density. This argument gives the optimal rate in
, showing the distance
between the
-particle
marginal density and the tensor product of the mean-field limit is
. We use
cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an
arbitrary order
,
these provide “low-dimensional” approximations to the
-particle marginal
density with error
.
Keywords
propagation of chaos, mean-field interacting diffusions,
BBGKY hierarchy