@MartinHairer The uniqueness part is in Remark 2.22 after the local existence theorem from the paper: renormalising SPDEs in regularity structures (the theorem gives sufficient conditions for the existence only) doesn't hold in general, this leaves us the question of this post: is it really "crucial-interesting" to have a thorough study of local well-posedness of SPDE never tackled before? One possible outcome of local well-posedness is trying to prove global well-posedness (in space or time)

@MartinHairer RhysSteele, the examples, which actually are my (thesis) research that I am studying using both ways, are the stochastic Cahn-Hilliard (sub-critical for $d\leq 5$) and stochastic Kuramoto-Sivashinsky equations (sub-critical for $d\leq 7$), deterministic cases of these equations have only been studied-published (for Cahn-hilliard it's $d=1,2,3$, Kuramoto-sivashinsky $d=1$) so far, on the other hand $d=4$ for C-H or $d=2,3,4$ for KS where we use Debussche-DaPrato are not yet tackled, $d=5$ for C-H, $d\geq 5$ for KS requires regularity structures or para-controlled distrobutions

@RhysSteele from your question, do I understand that they are all treated? (A paper, that doesn't do it for a "given equation", instead that deals with a large class of equations using regularity structures)

@m7e 1. You mean $\phi_\epsilon = P_rh-\int_0^rP_{r-q}(\phi_\epsilon^3-3c_\epsilon\phi_\epsilon)dq+X_\epsilon(r)$ the solution is defined point wise (mild formulation)? How come this is true? 2. How to relate the equation verified by $u_\epsilon$, defined in the distribution sense, to the equation verified by $\phi_\epsilon$, defined pointwise?

Hi, kindly can you elaborate? Do you mean to redefine the heat semi-group so that $(-\Delta)^{1/2}$ is excluded or?