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Stochastic processes with finite size scale invariance

We present a theory of stochastic processes that are finite size scale invariant. Such processes are invariant under generalized dilations that operate on bounded ranges of scales and amplitudes. We recall here the theory of deterministic finite size scale invariance, and introduce an operator called Lamperti transform that makes equivalent generalized dilations and translations. This operator is then used to defined finite size scale invariant processes as image of stationary processes. The example of the Brownian motion is presented is some details to illustrate the definitions. We further extend the theory to the case of finite size scale invariant processes with stationary increments.