How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?

Question

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.

Answer 1

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.