Compensated fractional derivatives and stochastic evolution equations

We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is beta-Holder continuous for beta is an element of (1/3.1/2). From the point of view of the Rough Path Theory, stochastic integrals related to the solution of ordinary differential equations contain area-elements from a tensor space. Based on (compensated) fractional derivatives we are able to derive a second mild equation for these area components. We formulate sufficient conditions for the existence and uniqueness of a pathwise mild solution by using the Banach fixed point theorem provided that the coefficients of the system are sufficiently regular. (C) 2012 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.