## An equivalence of $$H_{-1}$$ norms for the simple exclusion process.(English)Zbl 1019.60091

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.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory

### Keywords:

exclusion process; hydrodynamics; resolvent norm
Full Text:

### References:

 [1] ANDJEL, E. D. (1982). Invariant measures for the zero range process. Ann. Probab. 10 525-547. · Zbl 0492.60096 [2] DE MASI, A. and FERRARI, P. (1985). Self-diffusion in one-dimensional lattice gases in the presence of an external field. J. Statist. Phy s. 38 603-613. · Zbl 0624.60117 [3] FERRARI, P. and FONTES, L. (1994). Current fluctuations for the asy mmetric simple exclusion process. Ann. Probab. 22 820-832. · Zbl 0806.60099 [4] KIPNIS, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408. · Zbl 0601.60098 [5] KIPNIS, C. and LANDIM, C. (1999). Scaling Limits of Interacting Particle Sy stems. Springer, New York. · Zbl 0927.60002 [6] KIPNIS, C., LANDIM, C. and OLLA, S. (1994). Hy drody namical limit for a nongradient sy stem: The generalized sy mmetric simple exclusion process. Comm. Pure Appl. Math. 47 1475-1545. · Zbl 0814.76003 [7] KIPNIS, C. and VARADHAN, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes. Comm. Math. Phy s. 104 1-19. · Zbl 0588.60058 [8] LANDIM, C. and YAU, H. T. (1997). Fluctuation-dissipation equation of asy mmetric simple exclusion processes. Probab. Theory Related Fields 108 321-356. · Zbl 0884.60092 [9] LIGGETT, T. M. (1985). Interacting Particle Sy stems. Springer, New York. [10] LIGGETT, T. M. (1999). Stochastic Particle Sy stems: Contact, Exclusion and Voter Models. Springer, New York. · Zbl 0949.60006 [11] SAADA, E. (1987). A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15 375-381. · Zbl 0617.60096 [12] SEPPÄLÄINEN, T. and SETHURAMAN, S. (2003). Transience of second-class particles and diffusive variance bounds for additive functionals of one dimensional exclusion processes. Ann. Probab. 31 148-169. · Zbl 1029.60083 [13] SETHURAMAN, S. (2000). Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28 277-302. · Zbl 1044.60017 [14] SETHURAMAN, S. (2001). On extremal measures for conservative particle sy stems. Ann. Inst. H. Poincaré Probab. Statist. 37 139-154. · Zbl 0981.60098 [15] SETHURAMAN, S., VARADHAN, S. R. S. and YAU, H. T. (2000). Diffusive limit of a tagged particle in asy mmetric simple exclusion processes. Comm. Pure Appl. Math. 53 972-1006. · Zbl 1029.60084 [16] SETHURAMAN, S. and XU, L. (1996). A central limit theorem for reversible exclusion and zerorange particle sy stems. Ann. Probab. 24 1842-1870. · Zbl 0872.60079 [17] VARADHAN, S. R. S. (1995). Self-diffusion of a tagged particle in equilibrium for asy mmetric mean zero random walk with simple exclusion. Ann. Inst. H. Poincaré Probab. Statist. 31 273-285. · Zbl 0816.60093 [18] AMES, IOWA 50011 E-MAIL: sethuram@iastate.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.