Determine the stationary distribution of a non-dominant favorable site

Determine the full stationary distribution of the population fraction of a favorable site in the localized regime of the mean-field growth model with quenched heterogeneous growth rates and multiplicative temporal noise, conditional on the site not being the currently localized dominant site.

Background

In the localized regime, several favorable sites may compete over time, although typically only one carries an order-one population fraction at a given time. The authors describe the dominant site's distribution and propose an inverse-gamma ansatz for favorable sites that are not currently dominant.

The unresolved issue is the complete stationary law of a favorable site across both its localized and delocalized episodes. Resolving it would determine the conditional delocalized distribution and clarify the statistics of switching among favorable sites.

References

While we know that the distribution of a good site is given by $P_{\max}$ whenever it is localized, we do not yet know its full stationary distribution.

— Fluctuations and multifractality in stochastic models of interface growth and population dynamics  (2609.34468 - Bernard, 28 Sep 2026) in Section “Several favorable sites: an effective description in terms of $p_{\max}$,” Chapter “Mean-Field Growth with Quenched Heterogeneity and Multiplicative Noise”