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

Let u α , d = u α , d t , x , t ∈ 0 , T , x ∈ ℝ d be the solution to the stochastic heat equations (SHEs) with spatially colored noise. We study the realized power variations for the process u α , 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.

Limiting dynamics for stochastic nonclassical diffusion equations

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.</p>