Equilibrium Fluctuations of Reversible Dynamics

In Chapters 4 to 7 we examined the hydrodynamic behavior of several mean-zero interacting particle systems and proved a law of large numbers under diffusive resealing for the empirical measure. We now investigate the fluctuations of the empirical measure around the hydrodynamic limit starting from an equilibrium state. To fix ideas, we consider the nearest neighbor symmetric zero range process. The reader shall notice, however, that the approach presented below applies to a large class of reversible models including nongradient systems. The generator of this process is 0.1 $$\left( {{L_N}f} \right)\left( \eta \right) = \sum\limits_{x,y \in {\Bbb T}_N^d} {p\left( y \right)g\left( {\eta \left( x \right)} \right)\left[ {f\left( {{\eta ^{x,x + y}}} \right) - f\left( \eta \right)} \right]} ,$$ where p(y) = 1/2 if |y| = 1 and 0 otherwise and g is a rate function satisfying the assumptions of Definition 2.3.1.