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.
The aim of this paper is to investigate the large-time asymptotic behavior of heat and fractional heat equations in two related discrete settings: the whole lattice Z d and finite connected subgraphs with Dirichlet exterior condition.
Let d ≥ 1 and s ∈ (0, 1]. On Z d , for initial data u 0 ∈ ℓ 1 (Z d ), we consider the Cauchy problem ∂ t u(t, x) + (-∆) s u(t, x) = 0, t > 0, x ∈ Z d , u(0, x) = u 0 (x),
x ∈ Z d .
(1.1)
Here (-∆) s (0 < s ≤ 1) denotes the fractional power of the discrete Laplacian on Z d , defined via Fourier multiplier
The Cauchy problem (1.1) admits a unique solution in ℓ 1 (Z d ), given by (see Section 2.1)
where G (s)
e -tω(ξ) s e i⟨x,ξ⟩ dξ, x ∈ Z d .
(1.2)
In parallel, let G = (V, E, µ, w) be a locally finite weighted connected graph, and let Ω ⊂ V be a finite connected subset. For s ∈ (0, 1], we denote by L D Ω,s the Dirichlet diffusion operator on Ω: for 0 < s < 1, L D Ω,s is the restricted Dirichlet fractional Laplacian (zero extension outside Ω, then restriction back to Ω); for s = 1, L D Ω,1 is the usual Dirichlet graph Laplacian, see Section 3.1. We study
x ∈ Ω.
(1.3)
In the following, we will show that the two models exhibit different asymptotic mechanisms. On Z d , decay is polynomial and governed by the lattice fractional heat kernel, with sharp first-order correction controlled by the first moment of u 0 . On finite subgraphs, the spectrum is discrete and the long-time profile is the first eigenvalue, with exponential convergence rate given by the spectral gap.
It is well known that diffusion phenomena are ubiquitous in natural sciences, and parabolic equations provide a fundamental mathematical framework to describe their evolution. From the PDE viewpoint, one of the central questions is the large-time asymptotic behavior : after long time, what profile dominates the solution, and at which rate does the convergence occur?
In the classical case (s = 1), the heat equation in R d ,
admits the convolution representation
t (x) = (4πt) -d/2 e -|x| 2 /(4t) .
Observe that integrating over all of R d , we obtain that the total mass of solutions is conserved for all time, that is,
for all t > 0, and u(t, •) converges to M G
(1) t in the classical self-similar scaling sense; equivalently, for every 1 ≤ p ≤ ∞,
whose solution is u(t, x) = P (s) t * u 0 (x), where P (s) t
is the fractional heat kernel (stable density), characterized by P (s) t (ξ) = e -t|ξ| 2s . Again, mass is conserved, and the large-time profile is M P (s) t . More precisely, for all 1 ≤ p ≤ ∞,
See, for instance, [Váz18].
A finer first-order asymptotic expansion is available under finite first absolute moment:
In that case one has the quantitative estimates 2s) , s ∈ (0, 1] and ∥u(t) -M P (s)
with corresponding L p -versions by interpolation. Moreover, the rate t -1/(2s) is optimal within this firstmoment class; see [Váz17;Váz18].
In bounded domains with homogeneous Dirichlet condition, the long-time behavior is no longer selfsimilar as in R d . Instead, it is governed by the spectral structure of the Dirichlet operator: the first mode dominates as t → ∞, and the remainder decays at an exponential rate determined by the spectral gap λ 2 -λ 1 . Hence the convergence to the principal profile is exponential, in sharp contrast with the algebraic decay rates in the whole-space setting. See, e.g., [CCR06;BSV14].
The Euclidean large-time theory has natural counterparts on curved spaces, where geometry can substantially modify both asymptotic profiles and decay mechanisms. A classical model case is H n , where the asymptotics differ from the Euclidean self-similar regime due to geometric effects [Váz22].
On complete noncompact manifolds, large-time asymptotics has been analyzed under suitable Ricci curvature assumptions, highlighting geometry as a decisive factor in the long-time regime [Li86; GPZ23]; for broader classes, notably noncompact Riemannian symmetric spaces, representation-theoretic methods yield sharper results for both heat and fractional heat equations, including quantitative L p -asymptotic estimates [APZ23; Pap24a; NRS24; Pap24b].
Compared with the Euclidean and manifold settings, large-time asymptotics for diffusion equations on graphs is still relatively less developed, especially for lattice graphs and nonlocal operators. Related analytical tools for the discrete Laplacian can be found in [Cia23]. For nonlocal discrete models, including the fractional discrete Laplacian together with regularity and applications, see [Cia+18]. For discrete evolutions, asymptotic questions have been addressed, for instance, in the study of the discrete-in-time heat equation [AA22]. On metric graphs, several works analyze diffusion asymptotics for both local and nonlocal models, including general diffusion problems and configurations with infinite edges, where effective reduced dynamics may emerge at large time [IRS21]. For the one-dimensional lattice Z, refined large-time behavior for the discrete hea
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