This paper studies the stochastic heat equation with multiplicative noises: ∂u ∂t = 1 2 ∆u + uẆ , whereẆ is a mean zero Gaussian noise and uẆ is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.2010 Mathematics Subject Classification. 60G15; 60H07; 60H10; 65C30. Key words and phrases. Fractional Brownian motion, Malliavin calculus, Skorohod integral, Young's integral, stochastic partial differential equations, Feynman-Kac formula, intermittency.This project has been carried out while S. Tindel was on sabbatical at the University of Kansas. He wishes to express his gratitude to this institution for its warm hospitality.

We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence.Comment: Published in at http://dx.doi.org/10.1214/08-AOP437 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail.