Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees

In this paper, we study the Fisher–KPP equation $${\mathrm{\partial }}_{t}u=\alpha {\mathrm{\Delta }}_{{\text{TT}}_{q+1}}u+\beta u(1-u),$$ on the homogeneous $(q+1)$-regular tree ${\text{TT}}_{q+1}$, with $q\ge 2$, for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold $$\beta >\alpha (\sqrt{q}-1{)}^2$$ and determine the associated critical speed $c_{\ast }$ and decay rate ${\lambda }_{\ast }$. In this regime, we prove that, for every fixed $\theta \in (0,1)$, the outermost $\theta$-level set satisfies $${\kappa }_{\theta }(t)=c_{\ast }t-\frac{3}{2{\lambda }_{\ast }}\log t+O(1),t\to \mathrm{\infty }\mathrm{.}$$ Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor $3\mathrm{/}2$ persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher–KPP models, to homogeneous trees.