Degree-two spectral-gap reduction for the Brownian energy process

Determine whether, for the Brownian energy process on an arbitrary weighted graph with arbitrary positive site weights, the spectral gap is always attained by a polynomial of degree at most two in the energy variables.

Background

The Brownian energy process (BEP) is a conservative diffusion of energies on a weighted graph with Dirichlet reversible measure, analogous in some respects to the KMP model. Existing results cited in the paper establish one-particle domination under the condition that all site weights are at least one, show that one-particle domination can fail outside that regime, and prove an asymptotic two-particle reduction under a small-parameter scaling.

The paper explains that its hidden-model method does not directly apply to BEP because the formal hidden operator is not Markovian. Consequently, the question of whether all positive site weights and arbitrary weighted graphs still admit a degree-two spectral-gap reduction remains unresolved.