Scaling limits for some P.D.E. systems with random initial conditions

Let X = {X(x, t), x ∈ ℝⁿ, t ∈ R₊} be the R²-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X₀ = (u₀, v₀). We discuss the scaling limit of X under suitable conditions on X₀. Since the component fields u, v are dependent, even when the initial data u₀, v₀ are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u₀, v₀) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2] for the single heat equation can be discussed for the two-equation system, in a significant way.