Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? Rd, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k-?u(kZt, kx) converges in some sense as k ? 8, 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-?u(kzt, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.
|Number of pages||20|
|Journal||Communications in Mathematical Physics|
|Publication status||Published - 2000|