Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees
In this paper, we study the Fisher--KPP equation \[ \partial_t u = αΔ_{\TT_{q+1}}u+βu(1-u), \] on the homogeneous \((q+1)\)-regular tree \(\TT_{q+1}\), with \(q\geq2\), 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 \[ β>α(\sqrt q-1)^2 \] and determine the associated critical speed \(c_\ast\) and decay rate \(λ_\ast\). In this regime, we prove that, for every fixed \(θ\in(0,1)\), the outermost \(θ\)-level set satisfies \[ κ_θ(t) = c_\ast t-\frac{3}{2λ_\ast}\log t+O(1), \qquad t\to\infty. \] Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor \(3/2\) 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.