An equivalence of $H_{-1}$ norms for the simple exclusion process

Abstract

Resolvent $H_{-1}$ norms with respect to simple exclusion processes play an important role in many problems with respect to additive functionals, tagged particles, and hydrodynamics, among other concerns. Here, general translation-invariant finite-range simple exclusion processes with and without a distinguished particle are considered. For the standard system of indistinguishable particles, it is proved that the corresponding $H_{-1}$ norms are equivalent, in a sense, to the $H_{-1}$ norms of a nearest-neighbor system. The same result holds for systems with a distinguished particle in dimensions $d\geq 2$. However, in dimension $d=1$, this equivalence does not hold. An application of the $H_{-1}$ norm equivalence to additive functional variances is also given.