Scaling Limits of Wick Ordered KPZ Equation

Abstract:

Consider the KPZ equation , x∈ℝ^d, where W(t,x) is a space-time white noise. This paper investigates the question of whether, for some exponents χ and z, k ^{−χ} u(k ^z t, kx) converges in some sense as $k\to\infty$, and if so, what are the values of these exponents. The non-linear term in the KPZ equation is interpreted as a Wick product and the equation is solved in a suitable space of stochastic distributions. The main tools for establishing the scaling properties of the solution are those of white noise analysis, in particular, the Wiener chaos expansion. A notion of convergence in law in the sense of Wiener chaos is formulated and convergence in this sense of k ^{−χ} u(k ^z t, kx) as k→∞ is established for various values of χ and z depending on the dimension d.