Abstract
This article is devoted to define and solve an evolution equation of the form dy t = Δy t dt + dX t (y t ), where Δ stands for the Laplace operator on a space of the form $${L^p(\mathbb R^n)}$$ , and X is a finite dimensional noisy nonlinearity whose typical form is given by $${X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}$$ , 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 $${L^p(\mathbb R^n)}$$ . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.
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