Asymptotic Distributions for Power Variation of the Solution to a Stochastic Heat Equation

Let u = {u(t,x),t ∈ [0,T],x ∈ ℝ} be a solution to a stochastic heat equation driven by a space-time white noise. We study that the realized power variation of the process u with respect to the time, properly normalized, has Gaussian asymptotic distributions. In particular, we study the realized power variation of the process u with respect to the time converges weakly to Brownian motion.