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

TL;DR

This paper investigates whether the heat content on quantum graphs obeys a Rayleigh-Faber-Krahn inequality, proving it at small and large times using spectral theory and random walk methods, and developing new probabilistic expressions.

Contribution

It introduces a novel probabilistic approach to analyze heat content on quantum graphs and establishes the inequality at extremal times, extending classical spectral results.

Findings

Inequality holds at small and large times.

New expression for heat content via expected return times.

Spectral and probabilistic methods complement each other.

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.