Confinement of the infrared divergence for the Mumford process

Abstract

The Mumford process X is a stochastic distribution modulo constant and cannot be defined as a stochastic distribution invariant in law by dilations. We present two expansions of X—using wavelet bases—in X = X 0 + X 1 which allow us to confine the divergence on the “small term” X 1 and which respect the invariance in law by dyadic dilations of the process.