Confinement of the infrared divergence for the Mumford process

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=X0+X1 which allow us to confine the divergence on the “small term” X1 and which respect the invariance in law by dyadic dilations of the process.