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We are interested in parabolic differential equations $\partial_tu-a(u)\partial_x^2u=\xi$ with random forcing $\xi$, and non-linearity in the leading order term. For a forcing that is only in the (parabolic) Hölder space $C^\alpha$ with $\alpha<1$, this equation requires a renormalization. This can be done within Gubinelli's framework of controlled rough paths and, down to any $\alpha>0$, in Hairer's framework of regularity structures.

Within the setting of regularity structures, key ingredients are the identification of the structure and renormalization group. This is joint work with H. Weber, J. Sauer, S. Smith, and P. Linares.