In Martin Hairer's 2013 paper "Solving the KPZ equation", the process $X_\epsilon^\bullet$ is defined as the stationary solution to $$ \partial_t X_\epsilon^{\bullet} = \partial_x^2 X_\epsilon^{\bullet} + \Pi_0^{\perp} \xi_\epsilon $$ and then recursively, for each binary tree $\tau = [\tau_1, \tau_2]$ with $\bullet$ as the root, $$ \partial_t X_\epsilon^{\tau} = \partial_x^2 X_\epsilon^\tau + \Pi_0^{\perp} (\partial_x X_\epsilon^{\tau_1} \partial_x X_\epsilon^{\tau_2}) $$ with $\Pi_0^{\perp} = 1 - \Pi_0$, $\Pi_0$ being the orthogonal projection onto constants in $L^2$.

How to make sense of these equations, which include the product of solution derivatives and noise mollification? Can it be seen from the semigroup framework, as in Da Prato and Zabczyk's book, or is there a more appropiate point of view (like Hida's white noise approach)? Any reference would be greatly appreciated.