Long-run growth rate in a random multiplicative model

We consider the long-run growth rate of the average value of a random multiplicative process x_{i+1} = a_i x_iwhere the multipliers a_i=1+\rho\exp(\sigma W_i - \frac12 \sigma^2 t_i)have Markovian dependence given by the exponential of a standard Brownian motion W_i The average value \langle x_n\rangleis given by the grand partition function of a one-dimensional lattice gas with two-body linear attractive interactions placed in a...