An equivalence of \(H_{-1}\) norms for the simple exclusion process (Q1872323)

Resolvent \(H_{-1}\) norms with respect to simple exclusion processes play an important role in many problems with respect to additive functionals, tagged particles, hydrodynamics and so on. The author extends the norms to general (asymmetric) translation-invariant finite-range simple exclusion processes. As usual, the asymmetry costs a lot of problems. Based on a recent result by S. R. S. Varadhan, for the standard system of indistinguishable particles, the author proves that the corresponding resolvent \(H_{-1}\) norms are equivalent, in a sense, to the \(H_{-1}\) norms of a nearest-neighbor system. The same assertion is proved for systems with a distinguished particle in dimensions \(d\geq 2\), and 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.