### ON THE EVOLUTION OF GIBBS STATES OF THE LATTICE GRADIENT STOCHASTIC DYNAMICS OF INTERACTING OSCILLATORS

**W. I. SKRYPNIK**

*Theory of Stochastic Processes*

*Vol.15 (31), no.1, 2009, pp.61-82*

Grand canonical correlation functions of stochastic(Brownian) lattice linear oscillators interacting via a pair short-range potential are found in the thermodynamic limits at low activities and on a finite time interval. It is proved that their sequence is a weak solution of the BBGKY-type gradient diffision hierarchy. The initial correlation functions are Gibbsian, which corresponds to many-body positive finite-range and short-range non-positive pair interaction potentials. The utilized technique is based on an application of the Feynmanâ€“Kac formula for solutions of the Smoluchowski equation and a representation of the time-dependent correlation functions in terms of correlation functions of a Gibbs lattice oscillator path system with manybody interaction potentials.