Long-term behaviour conjecture for interacting reflected OU processes on graphs

Let $G=(V,E)$ be a graph with $|V|\geq 2$, let $X$ be the limiting reflected diffusion from Theorem~, and let $\nu(G)$ denote the principal eigenvalue of $G$. The parameters are $\alpha$ and $\beta$.

Long-term behaviour conjecture. 1) If $\alpha<0$ and $\alpha+\beta\nu(G)<0$, then the process $X$ is positive recurrent with the stationary distribution $\mu$ defined in Theorem~. 2) If $\alpha=0$, $\beta\leq 0$ and $|V|=2$, then the process $X$ is null recurrent. 3) If either $\alpha>0$, or $\alpha<0$ and $\alpha+\nu(G)\beta\geq 0$, then the process $X$ is transient.

This conjecture concerns the long-term behaviour of the limiting process for a graph with at least two vertices. For a single vertex, the limit is a reflected one-dimensional OU process, whose recurrence properties are known; when $\beta=0$, the multidimensional process consists of independent reflected OU processes. The interacting case is presented as the problem of determining the collective long-term behaviour.