Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation

In this work we consider the problem of pattern formation modeled by a one dimensional stochastic reactiondiffusion equation with time periodic coefficients. In particular, we apply Large Deviations methods to obtain lower bounds on the probability that certain evenly spaced patterns will develop. Our estimates are optimized when the number of interfaces scales as (epsilon root T)(-1), where epsilon is the length-scale and T is the time-scale. For large times T = rho |ln epsilon| our lower bound is of order exp(-epsilon(2 rho)), suggesting high likelihood for evenly spaced patterns whose number of interfaces is of order (epsilon root rho|ln epsilon vertical bar)(-1). Numerical simulations provide support to the idea that the more likely number of interfaces, even among unevenly spaced patterns, follows this law.