Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dyt = yt dt + d Xt (yt ), where stands for the Laplace operator on a space of the form L p(Rn), and X is a finite dimensional noisy nonlinearity whose typical form is given by Xt (φ) = ∑N i=1 xi t fi (φ), where each x = (x (1), . . . , x (N ) is a γ -Hölder function generating a rough path and each fi is a smooth enough function defined on L p(Rn). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.