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

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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 \[ \partial_t u+(-\Delta)^s u=0,\qquad u(0)=u_0\in \ell^1(\mathbb Z^d), \] 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.