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

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 ≥ 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.