Non-linear rough heat equations

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(ℝ n), and X is a finite dimensional noisy nonlinearity whose typical form is given by X t(φ) = Σ Ni=1 x itfi(φ), 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(ℝ n). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed. © 2011 Springer-Verlag.