Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dy t = Δ y t dt + dX t ( y t ), where Δ stands for the Laplace operator on a space of the form , and X is a finite dimensional noisy nonlinearity whose typical form is given by , where each x = ( x (1) , … , x ( N ) ) is a γ -Hölder function generating a rough path and each f i is a smooth enough function defined on . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.