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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This research is supported by the ANR Project ECRU (ANR-09-BLAN-0114-01/2).
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Deya, A., Gubinelli, M. & Tindel, S. Non-linear rough heat equations. Probab. Theory Relat. Fields 153, 97–147 (2012). https://doi.org/10.1007/s00440-011-0341-z
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DOI: https://doi.org/10.1007/s00440-011-0341-z