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

Rui Chen, Bo Li

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.

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 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 and , we consider on the Cauchy problem 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 -estimate. As consequences, under finite first moment we obtain the optimal decay rate \(t^{-1/(2s)}\) in -asymptotics (), and we prove sharpness by explicit shifted-kernel examples. Without moment assumptions, we still establish convergence in the full -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 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.
Paper Structure (11 sections, 14 theorems, 284 equations)

This paper contains 11 sections, 14 theorems, 284 equations.

Table of Contents

  1. Introduction and Main Results
  2. Fractional Diffusion on the Lattice
  3. Foundations of Discrete Operators
  4. Large-time Asymptotic Convergence
  5. Pointwise and l1 Bounds for Kernel Differences
  6. Sharpness of the Convergence with Finite First Moment
  7. Asymptotics under General l1 Class
  8. Fractional Dirichlet Diffusion on Finite Subgraphs
  9. Setting and Equation
  10. Large-time Asymptotic Convergence
  11. Positivity Improving of the Semigroup

Key Result

Proposition 1.1

Let $d\ge1$, $s\in(0,1]$. Then there exists $C=C(d,s)>0$ such that for all $t>0$, $x,y\in\mathbb Z^d$, where $G_t^{(s)}$ is defined in (gaussian).

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Proposition 1.10
  • ...and 20 more