Nonnegativity of the ABP response in dimensions three and higher

Prove that the dimensionless response kernel K_d^{\mathrm{ABP}}(z,\Omega) is nonnegative for every wave-number variable z and frequency variable \Omega in all spatial dimensions d\geq3.

Background

The paper defines the dimensionless ABP response kernel K_d^{\mathrm{ABP}}(z,\Omega) from the free-particle trajectory statistics and uses its sign to determine whether individual bath modes can transfer positive mechanical power to a moving weak probe. In two dimensions, the authors analytically identify a negative large-wave-number response, so positive-power modes exist.

For dimensions d\geq3, the authors derive positive asymptotic behavior in several limits and numerically find no negative response over the examined parameter domains. A large-d analysis also indicates that the minimum occurs at zero frequency and approaches a positive value as d increases. These results motivate, but do not prove, the conjectured global nonnegativity.