Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees

We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to \mathrm{\infty }$. Second, using them, we show that solutions with initial data in weighted ${\mathrm{\ell }}^1$ classes, asymptotically factorize in ${\mathrm{\ell }}^p$ norms, $p\in [1,\mathrm{\infty }]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion.