Differentiability of multiplicative processes related to branching random walks

A family of one-dimensional branching random walks indexed by an interval define a martingale taking values in the space of continuous functions. We propose a new approach to study the differentiability of the limit of this martingale. Under suitable conditions, this differentiability is obtained by assuming that the functions defining the martingale are differentiable only once; there is no loss of regularity. In this sense there is a progress with respect to the corresponding result of Biggins (1991). (C) 2000 Editions scientifiques et medicales Elsevier SAS.