In Martin Hairer's 2013 paper "[Solving the KPZ equation](https://annals.math.princeton.edu/wp-content/uploads/annals-v178-n2-p04-s.pdf)", 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](https://www.cambridge.org/core/books/stochastic-equations-in-infinite-dimensions/6218FF6506BE364F82E3CF534FAC2FC5), or is there a more appropiate point of view (like Hida's white noise approach)? Any reference would be greatly appreciated.



How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?
What I mean is that
$$
X\_\epsilon^\tau(t) = \int\_{-\infty}^t P\_{t-s} \Pi\_0^\perp (\partial\_x X\_\epsilon^{\tau\_1}(s)\, \partial\_x X\_\epsilon^{\tau\_2}(s))\,ds\;,
$$
where $P\_t$ denotes convolution with the heat kernel. The product appearing on the right is the usual pointwise product of two random variables with values in the space of continuous functions. The projection $\Pi\_0^\perp$ guarantees that the integral converges.


If you replace this product by the Wick product from white noise analysis you'll get a *different* process which will not lead to the physically relevant notion of solution. 


