Limits of rescaled renormalized diffusion matrices

Let $D\subset\mathbb R^d$ be the domain of a renormalization class $\mathcal W$, let $(c_k)_{k\geq0}$ be positive migration constants, and define $s_n=\sum_{k=0}^{n-1}c_k^{-1}$ and the iterated renormalizations $F^{(n)}w$. Let $\overline F_\gamma$ denote the rescaled renormalization transformation. Assume $s_n\to\infty$ and $s_{n+1}/s_n\to1+\gamma^*$ for some $\gamma^*\in[0,\infty]$. Limits of rescaled renormalized diffusion matrices. For every $w\in\mathcal W$, there is a limit $w^*$ such that

$$s_nF^{(n)}w\longrightarrow w^*.$$

Moreover, $w^*$ satisfies $\overline F_{\gamma^*}w^*=w^*$ if $0<\gamma^*<\infty$, satisfies

$$\frac12\sum_{i,j=1}^dw^*_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j}w^*(x)+w^*(x)=0\qquad(x\in\overline D)$$

if $\gamma^*=0$, and satisfies $\lim_{\gamma\to\infty}\overline F_\gamma w^*=w^*$ if $\gamma^*=\infty$. The conjecture is proved in the cited context for catalytic Wright–Fisher diffusions when $\gamma^*<\infty$, while the general case, especially $\gamma^*=0$, remains open.