  A local expansion is proposed for two-point distributions involving an
ultraviolet regularization in a four-dimensional globally hyperbolic
space-time. The regularization is described by an infinite number of functions
which can be computed iteratively by solving transport equations along null
geodesics. We show that the Cauchy evolution preserves the regularized Hadamard
structure. The resulting regularized Hadamard expansion gives detailed and
explicit information on the global dynamics of the regularization effects.
