Gradient type noises II - Systems of stochastic partial differential equations, preprint 2008. 23 A. Kamont, On the fractional anisotropic Wiener

The present paper is the second and main part of a study of partial differential equa-tions under the influence of noisy perturbations. Existence and uniqueness of mild solutions are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems. If the noise data are random, the results may be seen as a pathwise approach to SPDE’s. For typical examples, such as spatially one-dimensional stochastic heat equations with additive or multiplicative perturbations of fractional Brownian type, we recover and extend known results. In addition, we propose to consider partial noises of low order that allow to obtain function solutions in any space dimension. Key words: Stochastic partial differential equations, fractional Brownian sheet, fractional calculus, semigroups, function spaces 1991 MSC: 26A33, 35K60, 47D06, 35K60