Large-Time Asymptotics for Heat and Fractional Heat Equations on the Lattice and General Finite Subgraphs

Abstract: In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice (\mathbb Z^d) and finite connected subgraphs with Dirichlet boundary condition. These results provide a unified discrete theory of long-time asymptotics for local and nonlocal diffusions. For (d\ge1) and (s\in(0,1]), we consider on (\mathbb Z^d) the Cauchy problem [ \partial_t u+(-Δ)^s u=0,\qquad u(0)=u_0\in \ell^1(\mathbb Z^d), ] and derive a precise first-order asymptotic expansion toward the lattice fractional heat kernel (G_t^{(s)}). The main technical input is a pair of sharp translation-increment bounds for (G_t^{(s)}): a pointwise estimate and an (\ell^1)-estimate. As consequences, under finite first moment we obtain the optimal decay rate (t^{-1/(2s)}) in (\ell^p)-asymptotics ((1\le p\le\infty)), and we prove sharpness by explicit shifted-kernel examples. Without moment assumptions, we still establish convergence in the full (\ell^1)-class, and we show that no universal quantitative rate can hold in general. We also analyze fractional Dirichlet diffusion on finite connected subgraphs (restricted fractional setting, including (s=1) as the local case). In this finite-dimensional framework, solutions admit spectral decomposition and exhibit exponential large-time behavior governed by the principal eigenvalue and the spectral gap. In addition, we study positivity improving properties of the associated semigroups for both the lattice and Dirichlet evolutions.