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@MartinHairer: Thanks for pointing this out, I overlooked the effect of that nonlinearity. I proposed the substitution mostly to enforce nonnegativity. I have now two questions that might still help towards an answer: i) is it true that solutions of the equation without nonlinear part are a.e. nonnegative (as I would expect), and ii) can the formulation of X. Mao be extended to Hilbert space setting (in Da Prato & Zabczyk I encounter a weaker variant). This could work since $F[u](x) := -u^2(x) \mathbb 1_{u(x) \geq 0}$ still has this 'dissipativity' property. I'll look into this.
Thanks a lot for your comment. I was not aware of the notion of skew-symmetric matrices, so that helps. But I have some trouble understanding the concept of biregular graph in this context. To start with, we need a partition of the graph represented by $\Gamma$ into two sets of vertices. What would these partitions be, e.g. in the simplest example, where $\Gamma = \begin{pmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{pmatrix}$ representing a cycle over three vertices? I'll have a look at the references you mention (thanks!) but these may be a tough read for me with my analysis background.
I am thinking of the adjacency matrix of a graph as a matrix $A$ where $a_{ij} = 1$ whenever there is an edge between vertices $i$ and $j$, and zero otherwise. Is this not the usual definition of adjacency matrix? (I am not an expert on graph theory.) So (iii) requires $\Gamma(i,j) = 0$ for all $(i, j)$ such that $(i j)$ is not an edge of $G$. Note that $G$ is not a directed graph (in the formulation above).
