So, this is a theorem that I need to apply on a simpler system, and understanding it thus far has been the bane of my existence. I was wondering how exactly both Ito's formula and the Burkholder-Davies-Gundy Theorem are applied in each case. I think that so far I have not been able to mentally conquer Ito's Theorem. It makes sense to me where the later terms come from in the proof, but not the first ones. Also, I am struggling to see how the theorem is applied.
An important definition in this proof is the following,
\begin{equation} \mathcal{Z}_{k_1} = \alpha_k \epsilon^{-1} \int^T_0 e^{-\epsilon^{-2}\lambda_k(T-\tau) }d\tilde{\beta}_\tau,\end{equation} where $\alpha_k$ is a coefficient related to damping, $\lambda_k$ is a complex eigenvalue of the main SPDE that this paper is looking at, and $\epsilon$ is a small parameter. $\tilde{\beta}_\tau$ is a rescaled Brownian motion.
The theorem is as follows! Any help would be appreciated.
