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Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion

Patrizio Bifulco, Matthias Täufer

TL;DR

The paper studies the heat content on quantum graphs and asks whether a Rayleigh–Faber–Krahn type inequality holds for graphs of fixed volume. It develops a dual approach: a probabilistic Feynman–Kac framework with Brownian motion on metric graphs to obtain a new positive, path-expansion representation of the heat content in terms of expected return times of discrete random walks, used to prove a small-time extremal property; and a spectral Mercer's theorem-based analysis to prove the large-time extremal property via eigenvalue comparisons, showing that path graphs maximize heat content at both ends of the time scale. The main results establish that the heat content satisfies the Faber–Krahn inequality for large times unconditionally and for small times under rational dependence of edge lengths, with equality only for path graphs; the all-times question remains open. These findings advance shape optimization on metric graphs, linking stochastic representations and spectral methods to understand diffusion energetics on networks.

Abstract

We study the heat content on quantum graphs and investigate whether an analogon of the Rayleigh-Faber-Krahn inequality holds. This means that heat content at time $T$ among graphs of equal volume would be maximized by intervals (the graph analogon of balls as in the classic Rayleigh-Faber-Krahn inequality). We prove that this holds at extremal times, that is at small and at large times. For this, we employ two complementary approaches: In the large time regime, we rely on a spectral-theoretic approach, using Mercer's theorem whereas the small-time regime is dealt with by a random walk approach using the Feynman-Kac formula and Brownian motions on metric graphs. In particular, in proving the latter, we develop a new expression for the heat content as a positive linear combination of expected return times of (discrete) random walks - a formulation which seems to yield additional insights compared to previously available methods such as the celebrated Roth formula and which is crucial for our proof. The question whether a Rayleigh-Faber-Krahn inequality for the heat content on metric graphs holds at all times remains open.

Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion

TL;DR

The paper studies the heat content on quantum graphs and asks whether a Rayleigh–Faber–Krahn type inequality holds for graphs of fixed volume. It develops a dual approach: a probabilistic Feynman–Kac framework with Brownian motion on metric graphs to obtain a new positive, path-expansion representation of the heat content in terms of expected return times of discrete random walks, used to prove a small-time extremal property; and a spectral Mercer's theorem-based analysis to prove the large-time extremal property via eigenvalue comparisons, showing that path graphs maximize heat content at both ends of the time scale. The main results establish that the heat content satisfies the Faber–Krahn inequality for large times unconditionally and for small times under rational dependence of edge lengths, with equality only for path graphs; the all-times question remains open. These findings advance shape optimization on metric graphs, linking stochastic representations and spectral methods to understand diffusion energetics on networks.

Abstract

We study the heat content on quantum graphs and investigate whether an analogon of the Rayleigh-Faber-Krahn inequality holds. This means that heat content at time among graphs of equal volume would be maximized by intervals (the graph analogon of balls as in the classic Rayleigh-Faber-Krahn inequality). We prove that this holds at extremal times, that is at small and at large times. For this, we employ two complementary approaches: In the large time regime, we rely on a spectral-theoretic approach, using Mercer's theorem whereas the small-time regime is dealt with by a random walk approach using the Feynman-Kac formula and Brownian motions on metric graphs. In particular, in proving the latter, we develop a new expression for the heat content as a positive linear combination of expected return times of (discrete) random walks - a formulation which seems to yield additional insights compared to previously available methods such as the celebrated Roth formula and which is crucial for our proof. The question whether a Rayleigh-Faber-Krahn inequality for the heat content on metric graphs holds at all times remains open.
Paper Structure (8 sections, 9 theorems, 71 equations, 1 figure)

This paper contains 8 sections, 9 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notions and main results
  3. Basic notions on metric graphs
  4. Dirichlet Laplacian and the heat content
  5. The extremal Faber--Krahn property and main results
  6. Feynman--Kac formula and Brownian motion on metric graphs
  7. Proof of the Faber--Krahn inequality for small times
  8. Mercer's theorem for the heat content and Faber--Krahn inequality for large times

Key Result

Theorem 2.1

Let $\mathcal{G}$ be a compact metric graph, and let $\emptyset \subsetneq \mathsf{V}_{\mathrm{D}} \subset \mathsf{V}$ such that $\mathcal{G} \setminus \mathsf{V}_{\mathrm{D}}$ is connected Then: Equality holds at sufficiently small or large times if and only if $\mathcal{G} = \mathcal{P}_{\vert \mathcal{G} \vert}$ with $\mathsf{v}_{\mathrm{D}}$ being either the vertex identified with $0$ or with

Figures (1)

  • Figure 1: Illustration of the choice of $k_0$ in the proof of Theorem \ref{['thm:main-thm']} in the case $k_0 = 6$. We compare the path graph on the right and a non-path graph on the left and consider the unique realization of a random walk of minimal length, originating from $\mathsf{v}_D$, going $k_0/2$ steps to the left and then returning to $\mathsf{v}_D$ in $k_0/2$ steps. On the graphs, all individual jumps have the same probability except for the $k_0/2$-th step. In it, the probability of jumping back is larger (namely $1/2$) in the path graph than it is in the other graph (in this case $1/3$).

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3: Extremal Faber--Krahn property
  • Theorem 2.1
  • Remark 3
  • Remark 4
  • Remark 5: A very cautious conjecture
  • Definition 4: Brownian motions on metric graphs
  • ...and 25 more