Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

Abstract

Let $u_{\alpha ,d}=\left\{u_{\alpha ,d}\left(t,x\right),\hspace{1em}t\in \left[0,T\right],x\in {ℝ}^d\right\}$ be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process $u_{\alpha ,d}$ , in time, having infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for SHEs with spatially colored noise. This work builds on the recent works on delicate analysis of variations of general Gaussian processes and SHEs driven by space-time white noise.