A Central Limit Theorem for a System of Interacting Particles

Abstract

Consider a system of particles moving at random in ℤ^d in such a way that its probability law is stationary in time and invariant under spatial shifts; denote by X(j,t) the number of particles at site j at time t, let ρ be its expectation. The corresponding fluctuation processes are the S’-valued processes N^ε, defined by

$$ \begin{array}{*{20}{c}} {{N^\varepsilon }\left( {\phi ,t} \right) = {\varepsilon ^{d/2}} \cdot \sum\limits_j {\phi \left( {\varepsilon j} \right)\left( {X\left( {j,t\mathop \varepsilon \nolimits^{ - 2} } \right) - \rho } \right),} }&{\phi \varepsilon S} \end{array} $$

((1))

(S’: space of Schwartz distributions on R^d, S: space of rapidly decaying smooth functions).