Abstract
In this paper, we study the central limit theorem for a perturbed stochastic heat equation in the whole space mathbb{R}^{d}, dge 1. This equation is driven by a Gaussian noise, which is white in time and correlated in space, and the differential operator is a fractional derivative operator. Burkholder’s inequality plays an important role in the proof.
Highlights
In this paper, we study the central limit theorem for a fractional stochastic heat equation in spatial dimension Rd driven by a spatially correlated noise: ⎧ ⎨ ∂ uε ∂t (t, x) = Dδα uε (t, x) + b(uε (t, x))√ εσ (uε (t, x))F (t, x), ⎩uε(0, x) = 0, (1)
1 Introduction In this paper, we study the central limit theorem for a fractional stochastic heat equation in spatial dimension Rd driven by a spatially correlated noise:
By Gronwall’s lemma, there exists a constant c(L, L , T) independent of ε such that sup E Xε(t, x) – X(t, x) 2 ≤ εc L, L , T
Summary
The study of the central limit theorem for stochastic (partial) differential equation has been carried out, see e.g. Where κ–δ and κ+δ are two non-negative constants satisfying κ–δ + κ+δ > 0 and φ is a smooth function for which the integral exists, and φ stands for its derivative. –ı ξ , x – t |ξi|αi exp i=1 π –ıδi 2 sgn(ξi) where ·, · stands for the inner product in Rd. 2.2 The driving noise F Let S(Rd+1) be the space of Schwartz test functions.
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