A LIMIT-THEOREM FOR CERTAIN DISORDERED RANDOM-SYSTEMS

Many disordered random systems in applications can be described by N r andomly coupled Ito stochastic differential equations in R(1): [GRAPHI CS] where (W-i)(i greater than or equal to 1) is a sequence of indepen dent copies of the one-dimensional Brownian motion W and (xi(i))(i gre ater than or equal to 1) is a sequence of independent copies of the R( p)-valued random vector xi. We show that under suitable conditions on the functions b, sigma, K and Phi the dynamical behaviour of this syst em in the N --> proportional to limit can be described by the non-line ar stochastic differential equation dX(t) = b(X(t)) dt + integral(R rh o+1) K(xi, x)Phi(X(t), y)P(t, dx dy) dt + sigma(X(t)) dW(t) where P(t, dx dy) is the joint probability law of xi and X(t).