Algebraic normalisation of regularity structures: can there be a explicit expression of g? This is related to the paper in the link: https://arxiv.org/abs/1610.08468 titled "Algebraic normalisation of regularity structures". In the method of re-normalization the functional $g$ shown in page 6 plays a major role. However, in the paper there is no explicit expression of $g$ shown (as an example for particular PDE). Is there any reference where I can find such example? Any comment or suggestion?
 A: Indeed, later on in the paper "Algebraic renormalisation of regularity structures" at the renormalization section "6 Renormalisation of model" they go over the generic constructions for the regularity structures that do not need to be built by hand each time and give concrete functionals.
But also in the original work at "A theory of regularity structures" at section " Renormalisation group associated to the general algebraic structure", he gives concrete examples of those renormalization functionals in terms of derivatives of the kernel eg. at (8.21), (8.29) the approximating functionals $f_{x}^{(\epsilon)}$ and the functionals $f_{x}$ defined as the dual of $F_x$ giving the shifting operator
$$\Gamma_{xy} = F_x^{-1}\circ F_y\;,$$
These functionals act on the regularity structure elements $\tau$. For example, on monomials they act concretely $f_{x}^{\epsilon}(X^m)=(-x)^m=f_{x}(X^m)$ and on the "integrated" element $\mathcal{J}_k \tau$  by
$$f_x^{(\epsilon)}\bigl( \mathcal{J}_k \tau\bigr) =  -\int D_1^k K(x,z)\,\bigl(\Pi_x^{(\epsilon)}\tau\bigr)(z)\,dz\;.$$
$$f_x (\mathcal{J}_k \tau) = -\int_{\mathbb{R}^d} D^k K(x, y) (\Pi_x \tau)(dy).$$
(The $\mathcal{J}_k \tau$ is again an abstract regularity structure element that improves the regularity of $\tau$ and it is the placeholder for the kth coefficient in the series
$$\mathcal{J}(x) \tau = \sum_{|k|_s < \alpha + \beta} {X^k\over k!} \int_{\mathbb{R}^d} D_1^{k}K(x,z) \bigl(\Pi_x \tau\bigr)(dz)\;,$$
and here the kernel $K$ coming from the particular PDE you are working with.)
As explained in Remark 8.35, this choice/expression
$$\Pi^M_x \tau = \bigl(\Pi_x \otimes f_x) \Delta^M \tau\;$$
help with having a clean ansatz that is linear in the above kernels and thus makes taking limits $\epsilon\to 0$ a lot easier too.
