The local-mass asymptotics conjecture for spatial branching processes with self-interaction

Let $Z_n$ be the population measure at generation $n$, let $g\in C_c^+\left(\mathbb R^d\right)$, and write $\langle Z_n,g\rangle$ for its integral against $g$. Let $\gamma$ be the interaction parameter, and let $\mathcal N^d(m,\Sigma)$ denote the $d$-dimensional normal distribution with mean $m$ and covariance matrix $\Sigma$. Local-mass asymptotics conjecture. If $\gamma>0$ (attraction), then there exists a random variable $N\sim\mathcal N^d(0,2)$ such that, conditional on $N=x_0$,

$$\lim_{n\to\infty}2^{-n}\langle Z_n,g\rangle=\left\langle\left(\frac{\gamma}{\pi}\right)^{d/2}\exp\left(-\gamma|\cdot-x_0|^2\right),g\right\rangle\quad\mathrm{a.s.}$$

Also, there exists a random variable $M\sim\mathcal N^d\left(0,\left(2+\frac{1}{4\gamma^2}\right)\mathbf I_d\right)$, with corresponding law $\mathbb P$, such that

$$\lim_{n\to\infty}2^{-n}E\langle Z_n,g\rangle=\mathbb E\langle M,g\rangle\quad\mathrm{a.s.}$$

Here $\mathbf I_d$ is the $d$-dimensional unit matrix. If $\gamma<0$ (repulsion), then

$$\lim_{n\to\infty}2^{-n}\langle Z_n,g\rangle=\langle 1,g\rangle\quad\mathrm{a.s.}$$

The conjecture combines the asymptotic decomposition into a limiting center of mass and a branching Ornstein–Uhlenbeck process with the strong law for local mass. The stated monotonicity of $2+1/(4\gamma^2)$ suggests that stronger attraction gives a smaller variance for the limiting distribution; the claims themselves are presented as conjectural.