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

Wensheng Wang,Wang Liao,Xiaoying Chang,Mehmed Nurkanovic

doi:10.1155/2021/8208934

Wensheng Wang, Wang Liao + Show 2 more

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https://doi.org/10.1155/2021/8208934

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Abstract

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.

Highlights

It is known that (1) admits a unique mild solution if and only if d < 2 + α, and this mild solution is interpreted as the solution of the following integral equation:
Bezdek [11] investigated weak convergence of probability measures corresponding to the solution of (1) in d 1
Swanson [13] showed that the solutions of the stochastic heat equation (SHE) in (6) with ε σ 1, in time, have infinite quadratic variation and are not semimartingales and investigated central limit theorems (CLTs) for modifications of the quadratic variations of the solutions of the SHEs with white noise

Summary

Results

Let Xα,d(t) uα,d(t, x), where x ∈ R is fixed. We will first show the exact convergence rate of variance for the realized power variation of the process Xα,d. By (13), we have the following convergence in probability for the realized power variation of the process Xα,d. E CLT for the realized power variation of the process Xα,d is as follows. As n tends to infinity, where B {B(t), t ∈ [0, T]} is a BM independent of the process Xα,d, and the convergence is in the space D([0, T]) equipped with the Skorokhod topology. Comparing (15) and (9), we have that the realized power variations of the process Xα,d for α + 1 ≤ d < α + 2 share similar Gaussian asymptotic properties with those of BM. 􏽚 􏽚 φ(x)f(x − y)ψ(y)dxdy (2π)− d􏽚 Fφ(ξ)Fψ(ξ)μ(dξ)

Rd Rd

Observe that

Let us now introduce the filtration