Heat Semigroup on Graphs

• Heat semigroups on graphs are families of operators derived from the Laplacian, modeling the time evolution of heat diffusion on both finite and infinite graphs.
• The formulation employs integral kernel representations, resolvent techniques, and asymptotic expansions to analyze spectral and stochastic behavior.
• Applications extend to metric, weighted, and directed graphs, including hypergraphs, bridging combinatorial, geometric, and probabilistic perspectives.

A heat semigroup on a graph is the family of operators generated by the (negative) Laplacian—interpreted as time evolution according to the heat equation—on spaces of functions defined on the graph. This formulation is central to analysis on metric graphs, weighted and infinite graphs, directed graphs, and generalizes to higher-order combinatorial structures such as hypergraphs. The modern theory encompasses spectral and stochastic properties, kernel expansions, boundary conditions, and geometric/curvature inequalities, offering a bridge between discrete, combinatorial, and geometric analysis.

1. Metric Graphs, Schrödinger Operators, and Domain Specification

A finite metric graph $\Gamma$ comprises a vertex set $V$ and edge set $E=E_{\mathrm{int}}\cup E_{\mathrm{ex}}$, with each internal edge $e\in E_{\mathrm{int}}$ identified with $[0,l_e]$ and each external edge $e\in E_{\mathrm{ex}}$ with $[0,\infty)$. Function spaces relevant for analysis are

$$L^2(\Gamma)=\bigoplus_{e\in E}L^2(0,l_e),\qquad H^m(\Gamma)=\bigoplus_{e\in E}H^m(0,l_e).$$

For each edge, a smooth potential $V_e(x)$ is fixed: $V_e\in C^\infty[0,l_e]$ for $e\in E_{\mathrm{int}}$ and $V_e\in C_c^\infty[0,\infty)$ for $e\in E_{\mathrm{ex}}$. The (formal) Schrödinger operator acts as

$(H\psi)_e(x) = -\psi_e''(x) + V_e(x)\,\psi_e(x).$

Self-adjointness is achieved by imposing boundary conditions at each vertex, characterized via projectors $P$ and self-adjoint matrices $L$, leading to domains: $$\operatorname{Dom}(H_{P,L}) = \{\,\psi\in H^2(\Gamma):\ (P+L)\psi_v + P^\perp\psi'_v = 0 \text{ for all } v\in V\,\}.$$ Kirchhoff conditions (continuity and vertex current conservation) are a notable special case (Bolte et al., 2014).

2. Heat Semigroup as an Integral Operator and Resolvent Connection

For $t>0$, the heat semigroup $e^{-tH}$ admits an integral kernel representation: $(e^{-tH} f)(x) = \int_\Gamma K_H(t;x,y) f(y)\,dy,$ where $$K_H(t;x,y)\in C^\infty((0,\infty)\times\Gamma\times\Gamma)\cap L^\infty$$.

An equivalent construction proceeds via the resolvent $R_H(\lambda) = (H-\lambda)^{-1}$ as an integral operator, with Laplace inversion: $$e^{-tH} = \frac{1}{2\pi i} \int_\gamma e^{-t\lambda} R_H(\lambda)\,d\lambda,$$ where $\gamma$ encircles the spectrum $\sigma(H)$. The resolvent kernel can be made explicit, decomposing into free Green’s functions and terms reflecting the effect of the vertex boundary conditions and potentials (Bolte et al., 2014).

The general abstract setting for infinite, weighted graphs defines the graph Laplacian $\Delta$ by

$$(\Delta f)(x) = \frac{1}{m(x)}\sum_{y\sim x}w(x,y)(f(x)-f(y)),$$

with $w(x,y)$ symmetric edge weights and $m(x)$ positive vertex weights. The semigroup $e^{-t\Delta}$ is strongly continuous and contractive on $\ell^2(V,m)$, with integral kernel $H_G(x,y;t)$ determined by

$$(e^{-t\Delta}f)(x) = \sum_{y\in V} H_G(x,y;t)f(y)m(y).$$

This kernel is the unique bounded solution to $\partial_t u(x,t) + \Delta_x u(x,t) = 0$, subject to the delta initial condition (Jorgenson et al., 2024).

3. Asymptotic Expansions and Small-Time Behavior

For compact metric graphs (no external edges), the heat semigroup is trace class, leading to the heat trace

$$\Theta(t) = \operatorname{Tr} e^{-tH} = \int_\Gamma K_H(t;x,x)\,dx.$$

This admits a full asymptotic expansion for $t\to 0^+$: $\Theta(t) \sim \sum_{n=0}^\infty a_n t^{(n-1)/2}.$ The coefficients $a_0, a_1, a_2, a_3$ can be written in terms of the total internal length $\mathcal{L}$, the limiting scattering matrix $S_\infty$, vertex data $L$, and integrals involving edge potentials. For non-compact graphs, a reference heat semigroup (Dirichlet or Neumann) is subtracted on the external edge "star graph," regularizing the trace and the expansion, with adjusted coefficients (Bolte et al., 2014).

On infinite weighted graphs, the parametrix (Minakshisundaram–Pleijel) approach provides a convergent series expansion for the heat kernel $H_G(x,y;t)$. For combinatorial distance $r=d_G(x,y)$: $H_G(x,y;t) = a_0(x,y)t^r + O(t^{r+1}),$ where $a_0(x,y)$ is an explicit sum over paths of minimal length, constructed from edge and vertex weights (Jorgenson et al., 2024).

4. Heat Kernel Construction and Gaussian-Type Estimates

Gaussian kernel approximations are foundational in both classical and discrete settings. On locally finite, weighted graphs with bounded degree and a distance $d(\cdot,\cdot)\ge \delta>0$, a "dilated Gaussian" kernel

$$H_a(x,y;t) = \frac{1}{\sqrt{m(x)m(y)}} \exp\left(-\frac{m(x)m(y)d(x,y)^2}{t}\right)$$

serves as an order-0 parametrix. The true heat kernel is then $H_G(x,y;t) = H_a(x,y;t) + (H_a * F_a)(x,y;t)$, with $F_a = O(t^0)$. This leads to two-sided Gaussian estimates

$$\frac{C_1}{m(x)}\exp\left(-c_1\frac{d(x,y)^2}{t}\right) \le H_G(x,y;t) \le \frac{C_2}{m(x)}\exp\left(-c_2\frac{d(x,y)^2}{t}\right),$$

valid under the metric doubling property (Jorgenson et al., 2024). In compact settings, combining the Davies–Gaffney–Grigor'yan lemma with the Li–Yau inequality further strengthens such bounds (Bauer et al., 2014).

5. Boundary Conditions, Subgraph Approximation, and Stochastic Completeness

The realization of heat semigroups with boundary conditions is crucial in both finite and infinite graphs. The Neumann Laplacian is defined via the energy form and its restriction to $\ell^2(X,m)$: $$\mathcal{E}(f,g) = \frac12 \sum_{x,y\in X}b(x,y)(f(x)-f(y))(g(x)-g(y)) + \sum_{x\in X} c(x) f(x)g(x).$$ The corresponding semigroup $P_t^{(N)} = e^{-t L^{(N)}}$ is positivity improving if the graph is connected, and can be constructed as an exhaustion limit of finite subgraph Neumann semigroups: $$P_{t,k}^{(N)} \rightarrow P_t^{(N)} \quad \text{strongly in } \ell^2(X,m).$$ Convergence to the Dirichlet semigroup occurs if and only if the Neumann and Dirichlet forms coincide (form uniqueness). Stochastic completeness (conservation of total mass) characterizes $\ell^1$-convergence in this exhaustion procedure (Keller et al., 2023).

An explicit edge condition: $b(x,y)\le C m(x)m(y),\ \forall x,y\in X$ guarantees the Feller property for the Neumann semigroup. For birth–death chains, classical criteria are given in terms of series involving the weights and measures (Keller et al., 2023).

6. Generalizations: Directed Graphs and Hypergraphs

The extension of heat semigroups to directed graphs and hypergraphs introduces new phenomena due to the breakdown of symmetry and Markov properties. The combinatorial Laplacian for a finite directed hypergraph $H = (V,E)$ with signed incidence matrix $\mathcal{I}$ is $\mathcal{L} = \mathcal{I}\,\mathcal{I}^\top$ (Mugnolo, 20 Oct 2025). The associated semigroup $S(t) = e^{-t\mathcal{L}}$ is self-adjoint and strongly continuous, but may lose positivity and stochasticity unless combinatorial conditions are met—specifically, off-diagonals must be nonpositive (Metzler matrix), and each hyperedge must be equipotent (balanced initial/terminal vertices).

Spectral representation in terms of eigenvalues and eigenvectors provides explicit decay laws and conditions for eventual positivity or contractivity. In generic directed hypergraphs, the spectral gap controls exponential stability, but positivity or $\ell^\infty$ contractivity typically fails except in highly constrained combinatorial regimes (Mugnolo, 20 Oct 2025).

For finite, strongly-connected directed graphs, the Chung Laplacian $\Delta$ is constructed using the stationary Perron measure and symmetrized transition kernels. The heat semigroup $P_t = \exp(t\Delta)$ propagates according to the Chung Laplacian and underlies gradient estimates and contractivity in (asymmetric) Wasserstein distance. Notably, lower Ricci curvature bounds are characterized via heat semigroup gradient decay and are equivalent to contraction properties along the flow (Ozawa et al., 2020).

7. Curvature, Heat Semigroups, and Functional Inequalities

In the context of discrete geometry, the heat semigroup reveals curvature properties via gradient and Harnack inequalities. The exponential curvature-dimension condition $\mathrm{CDE}(n,K)$ yields Li–Yau parabolic gradient bounds; combined with the Davies–Gaffney–Grigor'yan lemma, this leads to robust Gaussian two-sided bounds for the heat kernel on graphs, reflecting an interplay between combinatorial geometry and heat diffusion (Bauer et al., 2014).

For directed graphs, Ricci curvature can be formulated through Lin–Lu–Yau's notion, and equivalently characterized by the contraction rate of the heat semigroup in Lipschitz norm and transportation distance. When the curvature is positive, this translates into measure concentration results—quantifying the likelihood of large deviations for Lipschitz observables (Ozawa et al., 2020).

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References:

(Bolte et al., 2014) "Heat-kernel and resolvent asymptotics for Schrödinger operators on metric graphs" (Jorgenson et al., 2024) "Constructing heat kernels on infinite graphs" (Keller et al., 2023) "Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property" (Mugnolo, 20 Oct 2025) "The heat flow driven by the Laplacian of a directed hypergraph" (Bauer et al., 2014) "Davies-Gaffney-Grigor'yan Lemma on Graphs" (Ozawa et al., 2020) "Heat flow and concentration of measure on directed graphs with a lower Ricci curvature bound"