Scaling limits of solutions of the heat equation for singular non-Gaussian data
Let \(\{u(t,x), t\geq 0,x\in \mathbb{R}\}\) be the solution of the one-dimensional heat equation with an initial condition of the form \(G(\xi(x))\), where \(\xi(x)\) is a zero-mean stationary Gaussian process with variance 1 and \(G\) is a nonlinear function which satisfies \(E(G^2(\xi(0)))<\infty\). The authors show the weak convergence of the finite-dimensional distributions of the rescaled random field \({1\over B(\sqrt T)} [u(tT,x \sqrt T)-E(G (\xi(0)))]\), as \(T\) tends to infinity, to the finite-dimensional distributions of a random field \(Z_m(t,x)\), assuming that the correlation function \(B(| x|)\) of \(\xi(x)\) satisfies some suitable conditions. The random field \(Z_m(t,x)\) has a spectral representation in terms of multiple stochastic integrals of order \(m\geq 1\), where the integer \(m\), called the Hermitian rank of \(G\), is the first nonzero coefficient in the Chebyshev-Hermite series expansion of \(G-E(G(\xi(0)))\).
scientific article; zbMATH DE number 1279468